Potential-capacity and some applications

Abstract

We introduce a new capacity associated to a non negative function V. We apply this notion to the study of a necessary and sufficient condition to ensure the existence and uniqueness of a Schrödinger type equation with measure data and with an operator whose coefficients are discontinuous. Namely, for a potential V, f a bounded Radon measure on Ω, then the equation $L_{V} u = - Δ u + U \cdot \nabla u + V u = f$ has a solution in $\begin{matrix} L^{1} (V) \cap L_{0}^{1} (Ω) = {g measurable, \int_{Ω} | g | V d x is finite and lim_{ε \to 0} \int_{{x : dist (x; \partial Ω) ⩽ ε}} | g | d x = 0} \end{matrix}$ if and only if f does not charge “irregular points” of V, provided that the set of “irregular points” have a zero potential capacity. As a byproduct of our results, we have the non existence of a Green operator for some $L_{V}$ .

Our method is also based on a new topology and density of $C_{c}^{2} (Ω ∖ K)$ in $C_{0}^{2} (\overline{Ω})$ whenever K has a zero potential-capacity.

Keywords

Capacity Schrödinger equations potential measures

1. Introduction

In recent works (see [4,5]), we have studied the equation $\begin{matrix} - Δ ω + U \cdot \nabla ω + V ω = f \end{matrix}$ in a smooth open bounded domain Ω whenever the potential V is locally integrable on the domain, under the Dirichlet condition $u = 0$ on $\partial Ω$ .

The first natural question is: what happens if we remove this integrability condition on V?

When examining the prototype of V say $V (x) = | x - a |^{- m}$ , with $m > 0$ , $a \in Ω$ , we observe that there is an interaction between the point a, the power m and the right hand side f.

To describe the growth of V and such interaction, we introduce here a new capacity associated to the potential V. Roughly speaking, the more V contains “irregular points” the more its capacity will be small. In particular, we will focus on potential whose “irregular points” are of capacity zero.

This new capacity is slightly different to the usual one considered by many authors (see [12,17]) for a complete review). Indeed, we recall that, if K is a compact subset of an open set Ω of $R^{n}$ , then, for $1 ⩽ k < + \infty$ , $1 ⩽ q < + \infty$ the $W^{k, q}$ capacity of K is usually defined as $\begin{matrix} (1) & {Cap}_{W^{k, q}} (K) = inf {‖ φ ‖_{W^{k, q} (Ω)}^{q}, φ \in B_{K}^{k}} \end{matrix}$ where $\begin{matrix} (2) & B_{K}^{k} = {ψ \in C_{c}^{k} (Ω), 0 ⩽ ψ ⩽ 1, ψ = 1 in a neighborhood of K} . \end{matrix}$

Here, we shall consider a potential $V ⩾ 0$ on $Ω, V ≢ 0$ for $ψ \in C_{c}^{2} (Ω)$ , we define $\begin{matrix} (3) & ‖ ψ ‖_{V, \infty} = ‖ ψ ‖_{L^{1} (Ω)} + {‖ \frac{\nabla ψ}{\sqrt{V}} ‖}_{\infty} + {‖ \frac{Δ ψ}{V} ‖}_{\infty}, \end{matrix}$ and we shall associate, the following capacity function, for a compact K included in Ω $\begin{matrix} (4) & {Cap}_{V, \infty} (K) = inf {‖ ψ ‖_{V, \infty}, ψ \in B_{K}^{2}} \end{matrix}$

Such capacity possesses common properties as for the above classical capacities (see Section 2 below), namely, we will show in particular that $\begin{matrix} if {Cap}_{V, \infty} (K_{i}) = 0, i \in J (finite) then {Cap}_{V, \infty} (⋃_{i \in J} K_{i}) = 0 . \end{matrix}$ Roughly speaking, such capacity will measure how singular is the potential V? And how “large” is this singularity. For instance, if $a \in Ω$ and V behaves like $| x - a |^{- m}$ near a, then ${Cap}_{V, \infty} ({a}) = 0$ if $m ⩾ 2$ and ${Cap}_{V, \infty} ({a}) > 0$ if $m < 2$ . But one of the most important properties that we need for the applications are:

Theorem 1.
Let K be compact included in Ω. Assume that ${Cap}_{V, \infty} (K) = 0$ . Then there exists a sequence ${(ψ_{j})}_{j}$ , $ψ_{j} \in C_{c}^{2} (Ω)$ such that
$ψ_{j} (x) \to_{j \to + \infty} 0$ for a.e. in Ω and strongly in $L^{1} (Ω)$ , $ψ_{j} = 1$ on K. $\begin{matrix} \frac{| \nabla ψ_{j} |}{\sqrt{V}} \to_{j \to + \infty}^{} 0 and \frac{Δ ψ_{j}}{V} \to_{j \to + \infty}^{} 0 strongly in L^{\infty} (Ω) . \end{matrix}$

If furthermore $V \in L_{loc}^{\frac{n}{2}, 1} (Ω_{K})$ with $Ω_{K} = {x \in Ω, dist (x; K) > 0}$ , then, for all $x \in Ω - K$ $\begin{matrix} ψ_{j} (x) \to_{j \to + \infty}^{} 0, \end{matrix}$ more precisely, if $Ω_{K, 0} \subset \subset Ω_{K}$ , then $\begin{matrix} \underset{{\overline{Ω}}_{K, 0}}{Max} | ψ_{j} (x) | \to_{j \to + \infty}^{} 0, ‖ \nabla ψ_{j} ‖_{L^{n, 1} (Ω_{K, 0})} \to_{j \to + \infty}^{} 0 . \end{matrix}$

If $a \in Ω$ , $V (x) = | x - a |^{- m}$ then V is in $L^{\frac{n}{2}, 1} (Ω)$ if and only if $m < 2$ .

The natural question is then, can we give sufficient conditions to ensure that $\begin{matrix} (5) & {Cap}_{V, \infty} (K) = 0 ? \end{matrix}$ The answer to that question is naturally linked with the motivations of our study. One of them is the following:

Let $μ_{0}$ be the Dirac mass at the origin, m a positive parameter, then we observe the following phenomena:

If $m ⩾ 2$ then there is no solution of $\begin{matrix} (M_{1}) \{\begin{matrix} - Δ u (x) + \frac{u (x)}{| x |^{m}} = μ_{0} & in B (0; 1) \subset R^{n}, n ⩾ 2, \\ u (x) = 0 & if | x | = 1 . \end{matrix} \end{matrix}$ But if $m < 2$ , the above problem $(M_{1})$ possesses at least one solution u. The same phenomena were also given in [1].

Let us notice that $(M_{1})$ has a solution if $n = 1$ .

Another motivation that we shall prove in this note is the following removable type singularities result:
Proposition (removable singularities with potential).

Assume (for simplicity) that $V (x) = \sum_{i = 1}^{p} \frac{b_{i}}{| x - a_{i} |^{m_{i}}}$ , $m_{i} ⩾ 0$ , $b_{i} > 0$ , $a_{i} \in Ω$ and consider $K = {a_{i}, m_{i} ⩾ 2}$ , $w \in L^{1} (Ω; V) \cap L_{0}^{1} (Ω)$ such that $\forall φ \in C_{c}^{2} (Ω ∖ K)$ we have $\begin{matrix} \int_{Ω} w (- Δ φ + V φ) d x = 0 . \end{matrix}$ Then $\begin{matrix} w \equiv 0 . \end{matrix}$ Here $δ (x) = distance (x; \partial Ω)$ .

So, the natural question is that if we consider an arbitrary potential $V ⩾ 0$ , how can we replace the set of singularities $K = {a_{i}, m_{i} ⩾ 2}$ ?

The question seems to be linked with some density problem (with an adequate topology).

The tough problem linked with that question is the construction of an appropriate sequence smooth function vanishing over K and disappearing when we pass to the limit for an adequate topology. These are the purpose of our main results stated in the next section. Namely a generalization of the above proposition for a large class of potential V and applications to some existence and non existence result for weak or very weak solution. We shall provide few examples of compact K whose $(V, \infty)$ -capacity is zero.

2. Notations definitions – primary definitions and results

We shall keep the notation we used to employ. We set $\begin{matrix} L^{0} (Ω) = {v : Ω \to R Lebesgue measurable} \end{matrix}$ and we denote by $L^{p} (Ω)$ the usual Lebesgue space $1 ⩽ p ⩽ + \infty$ . Although it is not too often used, we shall use the notation $\begin{matrix} W^{1, p} (Ω) = W^{1} L^{p} (Ω) \end{matrix}$ for the associated Sobolev space. We need the following definitions:

Definition 1 (of the distribution function and monotone rearrangement).

Let $u \in L^{0} (Ω)$ . The distribution function of u is the decreasing function $\begin{array}{rcl} m = m_{u} : R & \to & [0, | Ω |] \\ t & \mapsto & measure {x : u (x) > t} = | {u > t} | . \end{array}$ The generalized inverse $u_{*}$ of m is defined by, for $s \in [0, | Ω | [$ , $\begin{matrix} u_{*} (s) = inf {t : | {u > t} | ⩽ s}, \end{matrix}$ and is called the decreasing rearrangement of u. We shall set $Ω_{*} =] 0, | Ω | [$ .

Definition 2.
Let $1 ⩽ p ⩽ + \infty$ , $0 < q ⩽ + \infty$ :
If $q < + \infty$ , one defines the following norm for $u \in L^{0} (Ω)$ $\begin{matrix} ‖ u ‖_{p, q} = ‖ u ‖_{L^{p, q}} : = {[\frac{1}{| Ω |} \int_{Ω_{}} {[t^{\frac{1}{p}} | u |_{ } (t)]}^{q} \frac{d t}{t}]}^{\frac{1}{q}} where | u |_{ } (t) = \frac{1}{t} \int_{0}^{t} | u |_{} (σ) d σ . \end{matrix}$

If $q = + \infty$ , $\begin{matrix} ‖ u ‖_{p, \infty} = sup_{0 < t ⩽ | Ω |} t^{\frac{1}{p}} | u |_{* } (t) . \end{matrix}$ The space $\begin{matrix} (6) & L^{p, q} (Ω) = {u \in L^{0} (Ω) : ‖ u ‖_{p, q} < + \infty} \end{matrix}$ is called a Lorentz space.

If $p = q = + \infty$ , $L^{\infty, \infty} (Ω) = L^{\infty} (Ω)$ .

The dual of $L^{1, 1} (Ω)$ is called $L_{exp} (Ω)$

Remark 1.
We recall that $L^{p, q} (Ω) \subset L^{p, p} (Ω) = L^{p} (Ω)$ for any $p > 1$ , $q ⩾ 1$ .
Definition 3.
If X is a Banach space in $L^{0} (Ω)$ , we shall denote the Sobolev space associated to X by $\begin{matrix} W^{1} X = {φ \in L^{1} (Ω) : \nabla φ \in X^{n}} \end{matrix}$ or more generally for $m ⩾ 1$ , $\begin{matrix} W^{m} X = {φ \in W^{1} X, \forall α = (α_{1}, \dots, α_{n}) \in N^{n}, | α | = α_{1} + \dots + α_{n} ⩽ m, D^{| α |} φ \in X} . \end{matrix}$ We also set $\begin{matrix} W_{0}^{1} X = W^{1} X \cap W_{0}^{1, 1} (Ω) . \end{matrix}$

We also need to recall the Hardy’s inequality in $L^{n^{'}, \infty}$ saying that if Ω is a bounded Lipschitz domain: $\begin{matrix} (7) & \int_{Ω} {| \frac{u}{δ} |}^{q} ⩽ c ‖ \nabla u ‖_{L^{n^{'}, \infty}}^{q} \forall u \in W_{0}^{1} L^{n^{'}, \infty} (Ω), \end{matrix}$ with $n^{'} = \frac{n}{n - 1}$ , $1 ⩽ q < n^{'}$ . This inequality can be obtained from the results of [14] (see also [6]) since $W_{0}^{1} L^{n^{'}, \infty} (Ω) \subset W_{0}^{1} L^{1} (Ω; 1 + | log δ |)$ .

We need the following Lemma whose proof is given in [9,16,17]
Lemma 2.1.
Let* $A \subset R^{n}$ be closed and for $x \in R^{n}$ let $d (x) = d (x; A)$ denote the distance from x to A. Let $\begin{matrix} U = {x : d (x) < 1} . \end{matrix}$ Then there is a function $ρ \in C^{\infty} (U - A)$ and a positive number $M = M (n)$ such that $\begin{array}{l} M^{- 1} d (x) ⩽ ρ (x) ⩽ M d (x), x \in U - A \\ | D^{α} ρ (x) | ⩽ c (α) d {(x)}^{1 - | α |}, x \in U - A, | α | = α_{1} + \dots + α_{n} . \end{array}$ In particular, the result holds if $A = \partial Ω$ boundary of an open bounded set Ω, in this case $\begin{matrix} ρ \in C^{\infty} (Ω) and d (x) = δ (x) = dist (x; \partial Ω) . \end{matrix}$
Definition 4 (of $(V, \infty)$ -capacity or potential-capacity).

Let $V ⩾ 0$ be a measurable function on Ω, V non identically zero, V is called a potential function.

The $(V, \infty)$ -capacity of a compact K included in Ω (or potential-capacity of K) is given by relation (4).

We will denote by c different constant, sometimes we will specify the dependence with respect to the data.

Property 1 (of $(V, \infty)$ -capacity).

For any compact K in Ω, we have

$measure (K) ⩽ {Cap}_{V, \infty} (K)$ .

If $K_{1}$ is another compact included in K then $\begin{matrix} {Cap}_{V, \infty} (K_{1}) ⩽ {Cap}_{V, \infty} (K) . \end{matrix}$

For all $ε > 0$ , there exists an open set ω containing K such that for all compact $K^{'}$ satisfying $K \subset K^{'} \subset ω$ , on has $\begin{matrix} {Cap}_{V, \infty} (K^{'}) ⩽ {Cap}_{V, \infty} (K) + ε . \end{matrix}$

If $V_{1}$ , $V_{2}$ are two nonnegative potential $V_{1} ⩽ V_{2}$ then $\begin{matrix} {Cap}_{V_{2}, \infty} (K) ⩽ {Cap}_{V_{1}, \infty} (K) . \end{matrix}$

Proof.

For $ψ \in B_{K}^{2}$ we have $measure (K) ⩽ \int_{Ω} ψ (x) d x ⩽ ‖ ψ ‖_{V, \infty}$ which gives the result.

If $K_{1} \subset K$ then $B_{K}^{2} \subset {B_{K}^{2}}_{1}$ . Therefore $\begin{matrix} {Cap}_{V, \infty} (K_{1}) ⩽ {Cap}_{V, \infty} (K) . \end{matrix}$

Let $ε > 0$ , then there exists $ψ_{ε} \in B_{K}^{2}$ such that $\begin{matrix} (8) & ‖ ψ_{ε} ‖_{V, \infty} - ε ⩽ {Cap}_{V, \infty} (K) ⩽ ‖ ψ_{ε} ‖_{V, \infty} . \end{matrix}$ Since $ψ_{ε} = 1$ in a neighborhood of K, thus there exists an open set of ω on which $ψ_{ε} = 1$ . Then for all compact $K^{'}$ with $K \subset K^{'} \subset ω$ one has $ψ_{ε} = 1$ on $K^{'}$ and then $ψ_{ε} \in {B_{K}^{2}}_{^{'}}$ . Thus, $\begin{matrix} {Cap}_{V, \infty} (K^{'}) ⩽ ‖ ψ_{ε} ‖_{V, \infty} ⩽ {Cap}_{V, \infty} (K) + ε . \end{matrix}$

If $0 ⩽ V_{1} ⩽ V_{2}$ then $\frac{1}{V_{2}^{α}} ⩽ \frac{1}{V_{1}^{α}}$ if $α = \frac{1}{2}$ , $α = 1$ from which we get the result.
□
Remark 2.

In the definition of $(V, \infty)$ -capacity, we can add a different power on the potential V but the choice of the power is linked with the applications.

The property (3) is the so-called continuity from the right in Choquet’s capacity theory.

Proof of Theorem 1.

As for relation (8) considering $ε = \frac{1}{j}$ , $j ⩾ 1$ we have a sequence ${(ψ_{j})}_{j}$ : $\begin{matrix} (9) & ψ_{j} = 1 on K, 0 ⩽ ‖ ψ_{j} ‖_{V, \infty} - {Cap}_{V, \infty} (K) ⩽ \frac{1}{j} . \end{matrix}$ If ${Cap}_{V, \infty} (K) = 0$ then $\begin{matrix} ‖ ψ_{j} ‖_{V, \infty} \to_{j \to + \infty}^{} 0 \end{matrix}$ which implies $\begin{matrix} {‖ \frac{\nabla ψ_{j}}{\sqrt{V}} ‖}_{\infty} \to_{j \to + \infty}^{} 0, {‖ \frac{Δ ψ_{j}}{V} ‖}_{\infty} \to_{j \to + \infty}^{} 0, and ‖ ψ_{j} ‖_{L^{1}} \to_{j \to + \infty}^{} 0 . \end{matrix}$ This last convergence implies that for a subsequence still denoted by $ψ_{j}$ that $\begin{matrix} ψ_{j} (x) \to_{j \to + \infty}^{} 0 for a.e. \end{matrix}$

Let $x \in Ω_{K}$ , there exists $r > 0$ so that $B (x; r) \subset Ω_{K}$ . From Poincaré–Sobolev’s inequality or P.D.E. regularity (see [7]) $\begin{array}{l} \begin{matrix} ‖ \nabla ψ_{j} ‖_{L^{n, 1} (B (x; r))} & ⩽ c [‖ ψ_{j} ‖_{L^{\frac{n}{2}, 1} (B (x; r))} + ‖ Δ ψ_{j} ‖_{L^{\frac{n}{2}, 1} (B (x; r))}] \\ ⩽ c [‖ ψ_{j} ‖_{L^{1} (Ω)} + {‖ \frac{Δ ψ_{j}}{V} ‖}_{\infty} ‖ V ‖_{L^{\frac{n}{2}, 1} (B (x; r))}] \to_{j \to + \infty}^{} 0, \end{matrix} \\ \underset{y \in \overline{B} (x; r)}{Max} | ψ_{j} (y) | ⩽ c ‖ \nabla ψ_{j} ‖_{L^{n, 1} (B (x; r))} + c ‖ ψ_{j} ‖_{L^{1} (Ω)} \to_{j \to + \infty}^{} 0 . \end{array}$ If $Ω_{K, 0}$ is open set relatively compact in Ω by recovering ${\overline{Ω}}_{K, 0}$ and using the same argument as the above result we deduce $\begin{matrix} \underset{y \in {\overline{Ω}}_{K, 0}}{Max} | ψ_{j} (y) | \to_{j \to + \infty}^{} 0, ‖ \nabla ψ_{j} ‖_{L^{n, 1} (Ω_{K, 0})} \to_{j \to + \infty}^{} 0 . \end{matrix}$

A direct and simple computation shows that $\begin{matrix} {‖ | x - a |^{- m} ‖}_{L^{\frac{n}{2}, 1}} < + \infty if and only if m < 2 . \end{matrix}$
□

3. Few examples of compact K having a $(V, \infty)$ -capacity zero

Theorem 2 (Comparison near a compact).

Let K be a compact in Ω, $V_{1}$ and $V_{2}$ two nonnegative potentials satisfying

$\exists η > 0$ such that $V_{1} ⩽ V_{2}$ on a compact set $\begin{matrix} K_{η} = {x \in Ω, d (x; K) \dot{=} dist (x; K) ⩽ 2 η} \subset Ω . \end{matrix}$

$V_{1}$ is bounded from below and above on $\begin{matrix} O_{η} = {x \in Ω : \frac{1}{2} η < d (x; K) < 2 η} i.e. 0 < \underset{O_{η}}{inf ess} V_{1} ⩽ \underset{O_{η}}{sup ess} V_{1} < + \infty . \end{matrix}$

Then there exists a constant

c_{η} > 0

such that

\begin{matrix} (10) & {Cap}_{V_{2}, \infty} (K) ⩽ c_{η} {Cap}_{V_{1}, \infty} (K) . \end{matrix}

In particular,

\begin{matrix} (11) & if {Cap}_{V_{1}, \infty} (K) = 0 then {Cap}_{V_{2}, \infty} (K) = 0 . \end{matrix}

Proof.
Let θ be in $C_{c}^{\infty} (Ω)$ such that $0 ⩽ θ ⩽ 1$ and $θ = 1$ on ${x \in Ω : d (x, K) ⩽ η}$ and $support (θ) \dot{=} supp (θ) \subset {x \in Ω : d (x; K) < \frac{3}{2} η}$ .

Let us show that there exists a constant $c_{θ} > 0$ such that for all $ψ \in B_{K}^{2}$ , $\begin{matrix} (12) & ‖ θ ψ ‖_{V_{2}, \infty} ⩽ c_{θ} ‖ ψ ‖_{V_{1}, \infty} \end{matrix}$ We need the following Lemma 3.1.
There exists a constant $c_{θ η} > 0$ such that for all $ψ \in B_{K}^{2}$ we have $\begin{matrix} (13) & | ψ (x) | ⩽ c_{θ η} [‖ ψ ‖_{L^{1} (Ω)} + ‖ \nabla ψ ‖_{L^{\infty} (O_{η})}] \forall x \in supp (θ) \cap {η ⩽ d (\cdot; K) ⩽ \frac{3}{2} η} . \end{matrix}$
Proof.
By the compactness of the set $\begin{matrix} H_{0} = supp (θ) \cap {η ⩽ d (\cdot; K) ⩽ \frac{3}{2} η} \end{matrix}$ we have a family ${(B (x_{i}; r_{i}))}_{i = 1, \dots, p}$ such that $\begin{matrix} H_{0} \subset ⋃_{i = 1}^{p} B (x_{i}; r_{i}) = O \subset O_{η} . \end{matrix}$ Applying the Sobolev embedding, we have a constant $c_{θ η} > 0$ $\begin{matrix} ‖ ψ ‖_{L^{\infty} (O)} ⩽ c_{θ η} [‖ ψ ‖_{L^{1} (Ω)} + ‖ \nabla ψ ‖_{L^{\infty} (O_{η})}], ψ \in B_{K}^{2} . \end{matrix}$ This gives the result. □

Let $ψ \in B_{K}^{2}$ then $‖ θ ψ ‖_{L^{1}} ⩽ ‖ ψ ‖_{L^{1}}$ . For $x \in Ω$ $\begin{matrix} \nabla (θ ψ) (x) = \nabla θ ψ (x) + θ (x) \nabla ψ (x) . \end{matrix}$ We distinguish 3 cases
If $x \notin supp (θ)$ then $\nabla (θ ψ) (x) = 0$ , therefore we have $\begin{matrix} \frac{\nabla (θ ψ)}{{\sqrt{V}}_{2}} (x) = 0 ⩽ ‖ ψ ‖_{V_{1}, \infty} . \end{matrix}$

If $x \in supp θ$ , $x \notin O_{η}$ then $V_{1} (x) ⩽ V_{2} (x)$ and $\begin{matrix} | \nabla (θ ψ) (x) | ⩽ | \nabla ψ (x) | : so that | \frac{\nabla (θ ψ)}{{\sqrt{V}}_{2}} (x) | ⩽ | \frac{\nabla ψ}{{\sqrt{V}}_{1}} (x) | ⩽ ‖ ψ ‖_{V_{1}, \infty} . \end{matrix}$

If $x \in supp θ$ , $x \in O_{η}$ we still have $V_{1} (x) ⩽ V_{2} (x)$ but $V_{1} (x) ⩾ {ess inf}_{O_{η}} V_{1} > 0$ so that

if $d (x; K) ⩽ η$ , $\nabla (θ ψ) (x) = \nabla ψ (x)$ so we still have $\begin{matrix} | \frac{\nabla (θ ψ)}{{\sqrt{V}}_{2}} (x) | ⩽ | \frac{\nabla ψ}{{\sqrt{V}}_{1}} (x) | ⩽ ‖ ψ ‖_{V_{1}, \infty}, \end{matrix}$

if $η < d (x; K) ⩽ \frac{3}{2} η$ , we use Lemma 3.1 to $\begin{matrix} | \frac{\nabla (θ ψ)}{{\sqrt{V}}_{2}} (x) | ⩽ c_{θ} [| ψ (x) | + \frac{| \nabla ψ |}{{\sqrt{V}}_{1}} (x)] ⩽ c_{θ η} ‖ ψ ‖_{V_{1, \infty}} . \end{matrix}$
Since $Δ (θ ψ) (x) = Δ θ (x) ψ (x) + 2 \nabla θ (x) \nabla ψ (x) + θ (x) Δ ψ (x)$ , we can argue as before to deduce that $\begin{matrix} (14) & | \frac{Δ (θ ψ)}{{\sqrt{V}}_{2}} (x) | ⩽ c_{θ η} ‖ ψ ‖_{V_{1, \infty}}, \forall x \in Ω . \end{matrix}$

We have shown $\begin{matrix} (15) & {‖ \frac{\nabla (θ ψ)}{{\sqrt{V}}_{2}} ‖}_{L^{\infty} (Ω)} + {‖ \frac{Δ (θ ψ)}{V_{2}} ‖}_{L^{\infty} (Ω)} ⩽ c_{θ η} ‖ ψ ‖_{V_{1, \infty}} . \end{matrix}$ Thus we deduce relation (12) $\forall ψ \in B_{K}^{2}$ . We then have $\begin{matrix} {Cap}_{V_{2, \infty}} (K) ⩽ c_{θ} {Cap}_{V_{1, \infty}} (K) . \end{matrix}$ □

Here are few examples of Theorem 2.
Corollary 2.1 (of Theorem 2).

Let A, be a closed set included in Ω whose measure is zero, $m > 2$ , $m \in R$ , V a potential such that there exists $η > 0$ , $c > 0$ with $V (x) ⩾ \frac{c}{d {(x; A)}^{m}}$ for $x \in {y : d (y; A) ⩽ 2 η}$ . Then $\begin{matrix} {Cap}_{V, \infty} (A) = 0 . \end{matrix}$

Proof.
Let us set $V_{1} (x) = \frac{c}{d {(x; A)}^{m}}$ , $x \in Ω$ . According to Theorem 2 it is sufficient to show that $\begin{matrix} {Cap}_{V_{1, \infty}} (A) = 0 . \end{matrix}$ Let $H \in C^{\infty} (R)$ such that $\begin{matrix} (16) & H \in C^{\infty} (R) such H (t) = \{\begin{matrix} 1 & if t ⩾ 2, \\ 0 & if t ⩽ 1 . \end{matrix} \end{matrix}$ and denote by $δ (x) = dist (x; \partial Ω)$ .

According to Lemma 2.1 that we have a function $ρ \in C^{\infty} (Ω)$ , two constants $c_{1} > 0$ , $c_{2} > 0$ such that
$c_{1} δ (x) ⩽ ρ ⩽ c_{2} δ (x)$ , $\forall x \in Ω$

$\forall α = (α_{1}, \dots, α_{n}) \in N^{n}$ , $\exists c_{α} > 0$ such that $\begin{matrix} | D^{α} ρ (x) | ⩽ c_{α} ρ {(x)}^{1 - | α |} with | α | = α_{1} + \dots + α_{n}, D^{α} = \frac{D^{α_{1} + \dots + α_{n}}}{\partial x_{1}^{α_{1}} \dots \partial x_{n}^{α_{n}}} . \end{matrix}$ More, we have $ρ_{A} \in C^{\infty} (Ω ∖ A)$ , $M = M (n) > 0$ , $c (α)$ , $| α | ⩽ 2$ such that for all $x \in Ω ∖ A$ ,

$M^{- 1} d (x; A) ⩽ ρ_{A} (x) ⩽ M d (x; A)$ ,

$| D^{α} ρ_{A} (x) | ⩽ c (α) d {(x; A)}^{1 - | α |}$ .
Consider the sequence $ψ_{j} (x) = (1 - H (j ρ_{A} (x))) H (j ρ (x))$ . Then $ψ_{j} \in C_{c}^{\infty} (Ω)$ and $j ⩾ j_{a}$ , large enough so that ${x \in Ω : ρ_{A} (x) < \frac{1}{j_{a}}} \subset {x \in Ω : dist (x; A) < dist (A; \partial Ω)}$ . $\begin{matrix} ψ_{j} (x) = \{\begin{matrix} 1 & if ρ_{A} (x) < \frac{1}{j}, \\ 0 & if ρ_{A} (x) ⩾ \frac{2}{j}, \\ 1 - H (j ρ_{A} (x)) & if \frac{1}{j} < ρ_{A} (x) < \frac{2}{j} . \end{matrix} \end{matrix}$ On the set $D_{j} = {\frac{1}{j} < ρ_{A} (x) < \frac{2}{j}}$ one has $\begin{matrix} \nabla ψ_{j} (x) = - j H^{'} (j ρ_{A} (x)) \nabla ρ_{A} (x), \end{matrix}$ so that $\begin{matrix} (17) & ρ_{A} (x) | \nabla ψ_{j} (x) | ⩽ c_{1} {‖ H^{'} ‖}_{\infty} j ρ_{A} (x) ⩽ c_{1 H} \end{matrix}$ and $\begin{matrix} Δ ψ_{j} (x) = - H^{″} (j ρ_{A} (x)) j^{2} {| \nabla ρ_{A} (x) |}^{2} - H^{'} (j ρ_{A} (x)) j Δ ρ_{A} (x) . \end{matrix}$

From which we have $\begin{matrix} (18) & ρ_{A} {(x)}^{2} | Δ ψ_{j} (x) | ⩽ c_{3} {‖ H^{″} ‖}_{\infty} {(j ρ_{A} (x))}^{2} + {‖ H^{'} ‖}_{\infty} c_{4} j ρ_{A} (x) ⩽ c_{2 H} . \end{matrix}$ Since the measure of A is zero and $ψ_{j} (x) \to_{j \to + \infty} 0$ $\forall x \in Ω ∖ A$ , we deduce by the Lebesgue dominated theorem that $\begin{matrix} ‖ ψ_{j} ‖_{L^{1}} \to_{j \to + \infty}^{} 0 . \end{matrix}$ Since $Δ ψ_{j} (x) = \nabla ψ_{j} (x) = 0$ outside of $D_{j}$ , we deduce from the above estimates $\begin{matrix} (19) & {‖ \frac{\nabla ψ_{j}}{{\sqrt{V}}_{1}} ‖}_{L^{\infty} (Ω)} ⩽ c_{1 H}^{'} \frac{1}{j^{\frac{m}{2} - 1}} \to_{j \to + \infty}^{} 0 \end{matrix}$ and $\begin{matrix} (20) & {‖ \frac{Δ ψ_{j}}{V_{1}} ‖}_{L^{\infty} (Ω)} ⩽ c_{2 H}^{'} \frac{1}{j^{m - 2}} \to_{j \to + \infty}^{} 0 . \end{matrix}$ Since $ψ_{j} \in B_{A}^{2}$ , we deduce $\begin{matrix} {Cap}_{V_{1}, \infty} (A) ⩽ ‖ ψ_{j} ‖_{V_{1}, \infty} \to_{j \to + \infty}^{} 0 . \end{matrix}$ □
Corollary 2.2 (of Theorem 2).

Let $S_{1} = {x \in R^{n} : | x | = 1}$ the unit sphere of $R^{n}$ , $m > 2$ assume that $S_{1} \subset Ω$ and let V a nonnegative potential such that there exists $η > 0$ , $c > 0$ such that $V (x) ⩾ \frac{c}{| | x | - 1 |^{m}}$ for all $x \in {y \in Ω, d (y; S_{1}) ⩽ 2 η}$ . Then $\begin{matrix} {Cap}_{V, \infty} (S_{1}) = 0 . \end{matrix}$

One important property concerns the potential-capacity of a finite union of compact $⋃_{i \in J} K_{i}$ such that ${Cap}_{V, \infty} (K_{i}) = 0$ we are not able to prove the subadditivity, but we also have:

Theorem 3.
Let V be a nonnegative potential, $K_{i}$ , $i \in J$ be a finite number of compact sets included in Ω. Assume that ${Cap}_{V, \infty} (K_{i}) = 0$ $\forall i \in J$ . Then $\begin{matrix} {Cap}_{V, \infty} (⋃_{i \in J} K_{i}) = 0 . \end{matrix}$
Proof.
Since ${Cap}_{V, \infty} (K_{i}) = 0$ there exists a sequence $ψ_{i j} \in C_{c}^{2} (Ω)$ such that for a.e. x, $\begin{matrix} ψ_{i j} (x) \to_{j \to + \infty}^{} 0, ‖ ψ_{i j} ‖_{V, \infty} \to_{j \to + \infty}^{} 0, \end{matrix}$ $ψ_{i j} (x) = 1$ in a neighborhood of $K_{i}$ , $0 ⩽ ψ_{i j} ⩽ 1$ . Let us consider $H \in C^{\infty} (R)$ , $0 ⩽ H ⩽ 1$ as in relation (16), $ρ \in C^{\infty} (Ω)$ equivalent to the distance function $δ (x) = dist (x; \partial Ω)$ .

Since $ψ_{i j} \in C_{c}^{2} (Ω)$ then we have a set $Δ_{i j} \subset {x \in \overline{Ω} : ψ_{i j} (x) = 0}$ which is an open neighborhood of the boundary. Therefore, we can consider the open set $Δ_{j} = ⋂_{i \in J} Δ_{i j}$ neighborhood of $\partial Ω$ .

Since $α_{j} = dist (Ω ∖ Δ_{j}; \partial Ω) > 0$ , we can consider a sequence $μ_{j} > 0$ , such that $μ_{j} < α_{j}$ and $μ_{j} \to 0$ as $j \to + \infty$ . One has, in this case, the set $\begin{matrix} {x \in Ω : ρ (x) ⩽ μ_{j}} \subset Δ_{j}, \end{matrix}$ otherwise, we will have a point x such $ρ (x) ⩽ μ_{j}$ and $x \in Ω ∖ Δ_{j}$ so that $\begin{matrix} dist (Ω ∖ Δ_{j}; \partial Ω) ⩽ ρ (x) . \end{matrix}$

The function $\begin{matrix} Φ_{j} (x) = (1 - \prod_{i \in J} (1 - ψ_{i j} (x))) H (\frac{2}{μ_{j}} ρ (x)) with 3 μ_{j} < dist (⋃_{i \in J} K_{i}; \partial Ω) \end{matrix}$ satisfies $\begin{array}{l} 1 . Φ_{j} (x) = 1, x \in ⋃_{i \in J} K_{i}, \\ 2 . Φ_{j} \in C_{c}^{2} (Ω), \\ 3 . 0 ⩽ Φ_{j} (x) ⩽ 1, Φ_{j} (x) = 1 - \prod_{i \in J} (1 - ψ_{i j} (x)) if ρ (x) > μ_{j}, \\ Φ_{j} (x) = 0 if ρ (x) ⩽ μ_{j} . \end{array}$ We shall set for simplicity $Φ_{i j} (x) = 1 - ψ_{i j} (x)$ .

For $x \in Ω$ such that $ρ (x) > μ_{j}$ , we have $H (\frac{2}{μ_{j}} ρ) = 1$ and $\begin{array}{l} (21) & \nabla Φ_{j} (x) = - \sum_{k \in J} \prod_{i \in J, k \neq i} Φ_{i j} (x) \nabla Φ_{k j} (x) \\ (22) & | \frac{\nabla Φ_{j}}{\sqrt{V}} (x) | ⩽ \sum_{k \in J} | \frac{\nabla Φ_{k j}}{\sqrt{V}} (x) | = \sum_{k \in J} | \frac{\nabla ψ_{k j}}{\sqrt{V}} (x) | ⩽ \sum_{k \in J} ‖ ψ_{k j} ‖_{V, \infty} . \end{array}$ We also has $\begin{array}{l} | Δ Φ_{j} (x) | ⩽ \sum_{k \in J} | Δ ψ_{k j} (x) | + \sum_{k \in J} \sum_{ℓ \in J} | \nabla ψ_{k j} (x) | | \nabla ψ_{ℓ j} (x) | \\ (23) & \begin{matrix} | \frac{Δ Φ_{j}}{V} (x) | & ⩽ \sum_{k \in J} | \frac{Δ ψ_{k j}}{V} (x) | + \sum_{(k, ℓ) \in J^{2}} | \frac{\nabla ψ_{k j}}{\sqrt{V}} (x) | | \frac{\nabla ψ_{ℓ j}}{\sqrt{V}} (x) | \\ ⩽ \sum_{k \in J} ‖ ψ_{k j} ‖_{V, \infty} + \sum_{(k, ℓ) \in J^{2}} ‖ ψ_{k j} ‖_{V, \infty} ‖ ψ_{ℓ j} ‖_{V, \infty} . \end{matrix} \end{array}$

If $ρ (x) ⩽ μ_{j}$ then $x \in Δ_{j}$ and $\begin{matrix} 1 - \prod_{i \in J} (1 - ψ_{i j} (x)) = 0 : Φ_{j} (x) = 0 . \end{matrix}$ We conclude that relations (22) and (23) hold true. Therefore, we always have $\begin{array}{l} (24) & {‖ \frac{\nabla Φ_{j}}{\sqrt{V}} ‖}_{L^{\infty} (Ω)} ⩽ \sum_{k \in J} ‖ ψ_{k j} ‖_{V, \infty}, \\ (25) & {‖ \frac{Δ Φ_{j}}{V} ‖}_{L^{\infty} (Ω)} ⩽ \sum_{k \in J} ‖ ψ_{k j} ‖_{V, \infty} + {(\sum_{k \in J} ‖ ψ_{k j} ‖_{V, \infty})}^{2} . \end{array}$ On other hand, by the Lebesgue dominated convergence theorem, we have $\begin{matrix} (26) & ‖ Φ_{j} ‖_{L^{1} (Ω)} \to_{j \to + \infty}^{} 0 . \end{matrix}$ Relations (24) to (26) yield that $\begin{matrix} ‖ Φ_{j} ‖_{V, \infty} \to_{j \to + \infty}^{} 0 . \end{matrix}$ Since $\begin{matrix} {Cap}_{V, \infty} (⋃_{i \in J} K_{i}) ⩽ ‖ Φ_{j} ‖_{V, \infty}, \end{matrix}$ we derive the result. □

As a consequence of Theorem 3, we have
Corollary 3.1 (of Theorem 3).

For $i \in {1, \dots, m}$ , let $a_{i} \in Ω$ , $r_{i} ⩾ 0$ , $m_{i} > 2$ , $c_{i} > 0$ real numbers.

Define $S_{i} = {x \in R^{n} : | x - a_{i} | = r_{i}}$ , $K = ⋃_{i = 1}^{m} S_{i}$ assumed to be included in Ω. Let V be a nonnegative potential such that there exists $η > 0$ such that $\begin{matrix} V (x) ⩾ \sum_{i = 1}^{m} \frac{c_{i}}{{(| x - a_{i} | - r_{i})}^{m_{i}}} on {y : dist (y; K) ⩽ η} . \end{matrix}$ Then $\begin{matrix} {Cap}_{V, \infty} (K) = 0 . \end{matrix}$

Proof.
We have seen in Corollary 2.1 of Theorem 2 that ${Cap}_{V, \infty} (S_{i}) = 0$ whenever $S_{i} \subset Ω$ . Applying Theorem 3, we deduce the result. □

In the above Corollaries 1 and 2 of Theorem 2 we may replace $S_{1}$ by any compact included in Ω whose measure is zero. Concrete examples for application are given in [3,13].

As we have announced in the introduction„ we have ${Cap}_{| x |^{- m}, \infty} ({0}) > 0$ if $m < 2$ . Here is the proof
Theorem 4.
Let V be a nonnegative potential, $a \in Ω$ be such that there exist $η > 0$ , $c > 0$ $\begin{matrix} V (x) ⩽ \frac{c}{| x - a |^{m}}, x \in B (a; 2 η) for some m < 2 . \end{matrix}$ Then $\begin{matrix} {Cap}_{V, \infty} ({a}) > 0 . \end{matrix}$
Proof.
Let us set $V_{1} (x) = \frac{c}{| x - a |^{m}}$ , $x \in Ω ∖ {a}$ .

Following Theorem 2, ${Cap}_{V, \infty} ({a}) ⩾ c_{η} {Cap}_{V_{1}, \infty} ({a})$ , $c_{η} > 0$ .

We have for $φ \in B_{{a}}^{2}$ , $\begin{array}{rcl} ‖ \nabla φ ‖_{L^{n, 1} (B (a; η))} & ⩽ & [‖ φ ‖_{L^{\frac{n}{2}, 1} (B (a; η))} + ‖ Δ φ ‖_{L^{\frac{n}{2}, 1} (B (a, η))}] \\ ⩽ & c ‖ φ ‖_{V_{1}, \infty} [1 + ‖ V_{1} ‖_{L^{\frac{η}{2}, 1} (B (a; η))}] \\ ⩽ & c_{1} ‖ φ ‖_{V_{1}, \infty} < + \infty . \end{array}$ Applying the Sobolev–Lorentz embedding $\begin{matrix} ‖ φ ‖_{L^{\infty} (B (a; η))} ⩽ c_{2} [‖ φ ‖_{L^{n, 1} (B (a; η))} + ‖ \nabla φ ‖_{L^{n, 1} (B (a; η))}] ⩽ c_{3} ‖ φ ‖_{V_{1}, \infty} . \end{matrix}$ Since $φ (x) = 1$ in a neighborhood of a, this last inequality implies $1 ⩽ c_{3} {Cap}_{V_{1}, \infty} ({a})$ , this implies the result. □
Remark 3.

As we state before, the choice of the power $\frac{1}{2}$ and 1 in the definition is linked with the application, it is clear we can use other power as $‖ \frac{\nabla ψ}{V^{α}} ‖$ and $‖ \frac{Δ ψ}{V^{β}} ‖$ , $α > 0$ , $β > 0$ (see [15]).

In Corollary 2.1 of Theorem 2, we may take $m = 2$ , but the proof to show that ${Cap}_{V, \infty} (A) = 0$ uses a different argument ([15] work in progress)

We define $\begin{matrix} C_{0}^{2} (\overline{Ω}) = {φ \in C^{2} (\overline{Ω}), φ = 0 on \partial Ω} . \end{matrix}$
4. Approximation of functions in $C_{0}^{2} (\overline{Ω})$

We shall introduce the following sets: $\begin{array}{l} L^{1} (Ω; V) \dot{=} L^{1} (V) = {g : Ω \to R measurable such that \int_{Ω} | g (x) | V (x) d x < + \infty} \\ L_{0}^{1} (Ω) = {g \in L^{1} (Ω) such that lim_{ε \to 0} \frac{1}{ε} \int_{{x : δ (x) ⩽ ε}} | g (x) | d x = 0} \end{array}$

Remark 4.
One has $\begin{matrix} L^{1} (Ω; δ) = {g \in L^{1} (Ω) : \int_{Ω} | g (x) | \frac{d x}{δ (x)} < + \infty} is strictly included in L_{0}^{1} (Ω) . \end{matrix}$ Indeed, it was shown in [14] that $\begin{matrix} if f ⩾ 0 f \in L^{1} (Ω; δ) ∖ L^{1} (Ω; δ (1 + | log δ |)) \end{matrix}$ then the unique solution $g \in L_{+}^{1} (Ω)$ of $\begin{matrix} - \int_{Ω} g Δ φ d x = \int_{Ω} φ f d x \forall φ \in C_{0}^{2} (\overline{Ω}) \end{matrix}$ verifies $\begin{matrix} \int_{Ω} \frac{g}{δ} (x) d x = + \infty . \end{matrix}$ But A. Ponce ([12], chap. 20) shows that we have $g \in L_{0}^{1} (Ω)$ .
Definition 5.
Let ϕ be in $C_{0}^{2} (\overline{Ω})$ . We will say that a sequence ${(φ_{j})}_{j}$ of $C_{0}^{2} (\overline{Ω})$ converges weakly in the sense of the potential V to ϕ if for all $g \in L^{1} (V) \cap L_{0}^{1} (Ω)$ : $\begin{array}{l} 1 . \int_{Ω} g φ_{j} V d x \to_{j \to + \infty}^{} \int_{Ω} g ϕ V d x \\ 2 . \int_{Ω} g \partial_{k} φ_{j} d x \to_{j \to + \infty}^{} \int_{Ω} g \partial_{k} ϕ d x, k = 1, \dots, n \\ 3 . \int_{Ω} g Δ φ_{j} \to_{j \to + \infty}^{} \int_{Ω} g Δ ϕ d x . \end{array}$ Here $\partial_{k}$ is the partial derivative with respect to the kth derivate.
Definition 6.
Let ϕ be in $C_{0}^{2} (\overline{Ω})$ . We will say that a sequence ${(φ_{j})}_{j}$ of $C_{0}^{2} (\overline{Ω})$ converges weakly-strongly in the sense of V to ϕ if for all $g \in L^{1} (V) \cap L_{0}^{1} (Ω)$ $\begin{matrix} 0 = lim_{j} \int_{Ω} | g | | φ_{j} - ϕ | V d x = lim_{j} \int_{Ω} | g | | \nabla φ_{j} - \nabla ϕ | d x = lim_{j} \int_{Ω} | g | | Δ φ_{j} - Δ ϕ | d x . \end{matrix}$

We have the
Theorem 5.
Let K be a compact in Ω and V a nonnegative potential. Assume that ${Cap}_{V, \infty} (K) = 0$ . Then $\begin{matrix} the set C_{c}^{2} (Ω ∖ K) is \underline{weakly-strongly} dense in C_{0}^{2} (\overline{Ω}) in the sense of the potential V . \end{matrix}$
Proof.
Let Φ be in $C_{0}^{2} (\overline{Ω})$ . Since ${Cap}_{V, \infty} (K) = 0$ one has a sequence ${(φ_{j})}_{j}$ , $φ_{j} \in C_{c}^{2} (Ω)$ such that $0 ⩽ φ_{j} ⩽ 1$ , $φ_{j} = 1$ in a neighborhood of K and $‖ φ_{j} ‖_{V, \infty} \to_{j \to + \infty} 0$ , $φ_{j} \to_{j \to + \infty} 0$ a.e. in Ω.

As before, we then have a sequence ${(μ_{j})}_{j}$ tending to zero $μ_{j} > 0$ such that $\begin{matrix} the set {x : δ (x) ⩽ μ_{j}} is included in {x : φ_{j} (x) = 0} . \end{matrix}$ Let H be the function given in (16).

Then the sequence $Φ_{j} = (1 - φ_{j}) H (\frac{2}{μ_{j}} ρ) Φ$ where ρ is the smooth function equivalent to the distance function δ with the conditions $\begin{matrix} ‖ \nabla ρ ‖_{\infty} < + \infty, ‖ ρ Δ ρ ‖_{\infty} < + \infty (see Lemma 2.1) \end{matrix}$ $Φ_{j}$ belongs to $C_{c}^{2} (Ω ∖ K)$ . We have the following pointwise relations $\begin{matrix} (27) & Φ_{j} (x) \to_{j \to + \infty}^{} Φ (x) a.e. in Ω . \end{matrix}$ If $ρ (x) > μ_{j}$ , we have: $\begin{matrix} (28) & \begin{array}{l} Φ_{j} (x) = (1 - φ_{j} (x)) Φ (x) \\ \nabla Φ_{j} (x) = - \nabla φ_{j} (x) Φ (x) + (1 - φ_{j} (x)) \nabla Φ (x) \\ Δ Φ_{j} (x) = - Δ φ_{j} (x) Φ (x) - 2 \nabla φ_{j} (x) \nabla Φ (x) + (1 - φ_{j} (x)) Δ Φ (x) . \end{array} \end{matrix}$ If $ρ (x) ⩽ μ_{j}$ , we know that $φ_{j} (x) = 0$ so that $\begin{matrix} (29) & Φ_{j} (x) = H (\frac{2}{μ_{j}} ρ (x)) Φ (x) if ρ (x) ⩾ \frac{1}{2} μ_{j} and Φ_{j} (x) = 0 otherwise. \end{matrix}$ Then on ${\frac{1}{2} μ_{j} ⩽ ρ ⩽ μ_{j}}$ , one has $\begin{matrix} \nabla Φ_{j} (x) = \frac{2}{μ_{j}} \nabla ρ H^{'} (\frac{2}{μ_{j}} ρ (x)) Φ (x) + H (\frac{2}{μ_{j}} ρ (x)) \nabla Φ (x) \end{matrix}$ and $\begin{array}{rcl} Δ Φ_{j} (x) & = & \frac{2}{μ_{j}} Δ ρ (x) Φ (x) H^{'} (\frac{2}{μ_{j}} ρ (x)) + {(\frac{2}{μ_{j}})}^{2} Φ (x) {| \nabla ρ (x) |}^{2} H^{^{″}} (\frac{2}{μ_{j}} ρ (x)) \\ (30) & + \frac{4}{μ_{j}} H^{'} (\frac{2}{μ_{j}} ρ (x)) \nabla ρ (x) \cdot \nabla Φ (x) + H (\frac{2}{μ_{j}} ρ (x)) Δ Φ (x) . \end{array}$ Let $g \in L^{1} (V) \cap L_{0}^{1} (Ω)$ . By the Lebesgue dominated theorem, we have $\begin{matrix} (31) & lim_{j \to + \infty} \int_{Ω} | g (x) | | Φ_{j} (x) - Φ (x) | V (x) d x = 0 \end{matrix}$ From relation (28), we derive $\begin{array}{rcl} \int_{{ρ > μ_{j}}} | \nabla Φ_{j} (x) - \nabla Φ (x) | | g (x) | d x & ⩽ & ‖ \nabla Φ ‖_{\infty} \int_{Ω} | φ_{j} (x) | | g (x) | d x \\ (32) & + ‖ φ_{j} ‖_{V, \infty} ‖ Φ ‖_{\infty} (\int_{Ω} | g (x) | V (x) d x) (\int_{Ω} | g | d x) . \end{array}$ (We have used the Cauchy Schwarz inequality: $\int_{Ω} | g | \sqrt{V} d x ⩽ (\int_{Ω} | g | V d x) (\int_{Ω} | g | d x)$ .)

Setting $A_{j} = {\frac{1}{2} μ_{j} ⩽ ρ ⩽ μ_{j}}$ , one has $\begin{matrix} (33) & \int_{{ρ ⩽ μ_{j}}} | g (x) | | \nabla Φ_{j} (x) - \nabla Φ (x) | d x ⩽ \int_{{ρ ⩽ μ_{j}}} | g (x) | | \nabla Φ (x) | d x + \int_{A_{j}} | g (x) | | \nabla Φ_{j} (x) | d x . \end{matrix}$ The first integral tends to zero using Lebesgue dominated theorem, while the second integral can be bound as $\begin{matrix} (34) & \begin{matrix} \int_{A_{j}} | g (x) | | \nabla Φ_{j} (x) | d x & ⩽ c_{Φ} [\frac{1}{μ_{j}} \int_{A_{j}} | g (x) | d x + \int_{A_{j}} | g (x) | d x] \\ \to_{j \to + \infty}^{} 0 since g \in L_{0}^{1} (Ω) . \end{matrix} \end{matrix}$ From relations (32) to (34) we derive $\begin{matrix} (35) & lim \int_{Ω} | g (x) | | \nabla Φ_{j} (x) - \nabla Φ (x) | d x = 0 . \end{matrix}$ Using the same argument as above, we have $\begin{matrix} (36) & lim_{j \to + \infty} \int_{Ω} | g (x) | | Δ Φ_{j} (x) - Δ Φ (x) | d x = 0 . \end{matrix}$ Indeed, we have $\begin{array}{rcl} \int_{{ρ ⩾ μ_{j}}} | g (x) | | Δ Φ_{j} (x) - Δ Φ (x) | d x & ⩽ & \int_{{ρ ⩾ μ_{j}}} | Δ φ_{j} | | Φ | | g | d x + \int_{{ρ ⩾ μ_{j}}} | φ_{j} | | Δ Φ | | g | d x \\ + 2 \int_{{ρ ⩾ μ_{j}}} | \nabla φ_{j} | | \nabla Φ | | g | d x \\ ⩽ & c_{Φ} ‖ φ_{j} ‖_{V, \infty} [\int_{Ω} | g | V d x] [1 + \int_{Ω} | g | d x] \\ (37) & + c_{Φ} \int_{Ω} | φ_{j} | | g | d x \to_{j \to + \infty}^{} 0 . \end{array}$ On ${ρ ⩽ μ_{j}}$ , we have: $\begin{matrix} (38) & \int_{{ρ ⩽ μ_{j}}} | g | | Δ Φ_{j} - Δ Φ | d x ⩽ \int_{{ρ ⩽ μ_{j}}} | g | | Δ Φ | + \int_{A_{j}} | g | | Δ Φ_{j} | d x . \end{matrix}$ The first term tends to zero, while for the last term we replace $Δ Φ_{j}$ by its expression: $\begin{matrix} (39) & \int_{A_{j}} | g | | Δ Φ_{j} | d x ⩽ I_{1 j} + I_{2 j} + I_{3 j} + I_{4 j} . \end{matrix}$ Using the fact that $| \frac{Φ (x)}{ρ (x)} | ⩽ c ‖ \nabla Φ ‖_{\infty}$ , and $| ρ Δ ρ (x) | ⩽ c_{2}$ for all $x \in Ω$ , we have: $\begin{array}{l} (40) & I_{1 j} ⩽ c \frac{1}{μ_{j}} \int_{A_{j}} | \frac{Φ (x)}{ρ (x)} | | Δ ρ (x) ρ (x) | | g (x) | d x ⩽ c \frac{1}{μ_{j}} \int_{A_{j}} | g (x) | d x, \\ (41) & I_{2 j} ⩽ c {(\frac{1}{μ_{j}})}^{2} \int_{A_{j}} | g (x) | ρ (x) d x ⩽ c \frac{1}{μ_{j}} \int_{A_{j}} | g (x) | d x, \\ (42) & I_{3 j} ⩽ c \frac{1}{μ_{j}} \int_{A_{j}} | g (x) | d x, I_{4 j} ⩽ c \int_{A_{j}} | g (x) | d x \end{array}$ Thus $\begin{matrix} (43) & \int_{A_{j}} | g | | Δ ϕ_{j} | d x ⩽ c [\frac{1}{μ_{j}} \int_{A_{j}} | g (x) | d x + \int_{A_{j}} | g (x) | d x], \end{matrix}$ the constant c is independent of j and g. From relations (37) to (43), we derive the result. □

One may also give sufficient conditions to ensure that a sequence converges weakly in the sense of V.

Here is an example of such result: Theorem 6.
Let ${(φ_{j})}_{j}$ be a sequence of $C_{0}^{2} (\overline{Ω})$ , K a compact in Ω, V a nonnegative potential such that V is upper semi-continuous, that is for all real t, the set ${V ⩾ t}$ is closed in Ω, and assume also that the set ${x : V (x) = + \infty}$ is of measure zero, and:
$‖ φ_{j} ‖_{V, \infty}$ remains bounded in $R_{+}$ ,

there exists $φ \in C_{0}^{2} (\overline{Ω})$ such that the sequence ${(φ_{j})}_{j}$ converges to φ in $L^{\infty} (Ω)$ -weak-star.
Then, ${(φ_{j})}_{j}$ converges weakly to φ in the sense of the potential V.
Sketch of the proof.
Let $1 > η > 0$ , the set ${x \in Ω : V (x) ⩾ \frac{1}{η}}$ is closed in Ω thus $Ω_{η} = {x \in Ω : \frac{1}{V} (x) > η}$ is open and we have a constant M such that $\forall j$ , $\forall η \in] 0, 1 [$ $\begin{matrix} (44) & ‖ \nabla φ_{j} ‖_{L^{\infty} (Ω_{η})} + ‖ Δ φ_{j} ‖_{L^{\infty} (Ω_{η})} ⩽ η^{- 1} ‖ φ_{j} ‖_{V, \infty} ⩽ M η^{- 1} . \end{matrix}$ Then we deduce that for all $η \in] 0, 1 [$ , $‖ φ_{j} - φ ‖_{C^{1} ({\overline{Ω}}_{η})} \to_{j \to + \infty} 0$ .

Since $Ω ∖ ⋃_{η > 0} Ω_{η}$ is of measure zero, therefore, $\begin{matrix} \frac{\nabla φ_{j}}{\sqrt{V}} ⇀ \frac{\nabla φ}{\sqrt{V}} and \frac{Δ φ_{j}}{\sqrt{V}} ⇀ \frac{Δ φ_{j}}{V} in L^{\infty} -weak-star when j \to + \infty . \end{matrix}$ From those convergences, we derive the result. □
An example of sequence satisfying Theorem 6.
As example, we can take A as in Corollary 2.1 of Theorem 2, $V (x) = d {(x; A)}^{- 2}$ and $ψ_{j} (x) = (1 - H (j ρ_{A} (x))) H (j ρ (x))$ , as in the proof of Corollary 2.1 of Theorem 2. Then, for $φ \in C_{0}^{2} (\overline{Ω})$ the sequence $φ_{j} (x) = (1 - ψ_{j} (x)) φ (x)$ satisfies conditions 1. and 2..

Indeed, since $φ_{j} (x) \to φ (x)$ $\forall x \in Ω ∖ A$ and $‖ φ_{j} ‖_{\infty} ⩽ ‖ φ ‖_{\infty}$ , we deduce that ${(φ_{j})}_{j}$ converges to φ in $L^{\infty} (Ω)$ -weakly-star.

The set ${x : V (x) = + \infty} = A$ is of measure zero and V is upper semi continuous.

More, $φ_{j} (x) = φ (x)$ if $ρ_{A} (x) ⩾ \frac{2}{j}$ , and $\nabla φ_{j} (x) = 0$ if $ρ_{A} (x) < \frac{1}{j}$ , on $D = {\frac{1}{j} < ρ_{A} < \frac{2}{j}}$ , we have $\begin{matrix} (45) & ρ_{A} (x) | \nabla φ_{j} (x) | ⩽ c_{1 φ} and ρ_{A} {(x)}^{2} | Δ φ_{j} (x) | ⩽ c_{2 φ} . \end{matrix}$ This implies $‖ φ_{j} ‖_{V, \infty} ⩽ M < + \infty$ . □
Corollary 6.1 (of Theorem 6).

Let $V (x) = d {(x; A)}^{- 2}$ , A a compact set of measure zero in Ω. Then $\begin{matrix} C_{c}^{2} (Ω ∖ A) is weakly dense in C_{0}^{2} (\overline{Ω}) in the sense of the potential V . \end{matrix}$

Proof.
Let $φ \in C_{0}^{2} (\overline{Ω})$ then $φ_{j} H (j ρ (x))$ is in $C_{c}^{2} (Ω ∖ A)$ and the above arguments imply the statement. □

5. Applications of the potential-capacity and the approximation of $C_{0}^{2} (\overline{Ω})$

As a first application of the above results, we shall prove a removable type problem.

Theorem 7.
Let K be compact included in Ω. Assume that $C_{c}^{2} (Ω ∖ K) = C_{c}^{2} (Ω_{K})$ is weakly dense in $C_{0}^{2} (\overline{Ω})$ in the sense of potential V and let $w \in L^{1} (Ω; V) \cap L_{0}^{1} (Ω)$ be such that for all $φ \in C_{c}^{2} (Ω_{K})$ we have $\begin{matrix} (46) & \int_{Ω} w (- Δ φ + V φ) d x = 0 . \end{matrix}$ Then w satisfies the same equation ( 46 ) with $φ \in C_{0}^{2} (\overline{Ω})$ .
Proof.
Let φ be in $C_{0}^{2} (\overline{Ω})$ . Then, we have a sequence ${(φ_{j})}_{j}$ , $φ_{j} \in C_{c}^{2} (Ω ∖ K)$ such that $\begin{array}{l} (47) & lim_{j \to_{\infty}} \int_{Ω} w φ_{j} V d x = \int_{Ω} w φ V d x . \\ (48) & lim_{j \to_{\infty}} \int_{Ω} w Δ φ_{j} d x = \int_{Ω} w Δ φ d x . \end{array}$ Since $0 = \int_{Ω} w (- Δ φ_{j} + V φ_{j}) d x$ thus we have the result by passing to the limit. □

Next, we recall the following Kato’s inequality (see [2,8,10,12]).
Lemma 5.1 (Kato’s inequality and weak maximum principle).

Assume that w and f are in $L^{1} (Ω)$ such $- Δ w = f$ in $D^{'} (Ω)$ . Then $\begin{array}{l} (49) & 1 . - Δ | w | ⩽ f sign (w) in D^{'} (Ω), \\ 2 . - Δ w_{+} ⩽ f {sign}_{+} (w) in D^{'} (Ω), \\ 3 . if - Δ w ⩽ 0 in C_{0}^{2} {(\overline{Ω})}^{'} (dual space) then w ⩽ 0, \\ {sign}_{+} (σ) = \{\begin{matrix} 1 & if σ > 0, \\ 0 & otherwise, \end{matrix} and sign (σ) = \{\begin{matrix} 1 & if σ > 0, \\ 0 & if σ = 0, \\ - 1 & if σ < 0 . \end{matrix} \end{array}$

Corollary 7.1 (of Theorem 7 and Lemma 5.1).

Under the same assumption as for Theorem 7 , the function w verifying relation ( 46 ) satisfies $\begin{matrix} w \equiv 0 . \end{matrix}$

Proof.
Since $D (Ω) = C_{c}^{\infty} (Ω) \subset C_{0}^{2} (\overline{Ω})$ , then following Theorem 7, we have $\begin{matrix} - Δ w = - V w \in L^{1} (Ω) in D^{'} (Ω) . \end{matrix}$ From Kato’s inequality, one has: $\begin{matrix} - Δ (| w |) ⩽ - V | w | ⩽ 0 . \end{matrix}$ Therefore, using the same arguments as for Theorem 7, the inequality holds in the dual space $C_{0}^{2} {(\overline{Ω})}^{'}$ , we conclude that $| w | ⩽ 0 : w = 0$ . □

Let V be a nonnegative potential and define the subset of Ω by $\begin{matrix} Ω_{V} = {x \in Ω, \exists r_{x} > 0 such that ‖ V ‖_{L^{\frac{n}{2}, 1} (B (x; r_{x}))} < + \infty} . \end{matrix}$ One can show that $Ω_{V}$ is an open set in Ω.

Thus its complement $K_{V} = Ω - Ω_{V}$ is a compact included in Ω.
Definition 7.
The points $K_{V}$ are called the irregular points of V.
Remark 5.
The choice of $K_{V}$ can be modified according to the application that one wants to do.

If $V (x) = | x - a |^{- m}$ , $a \in Ω$ and applying the first theorem, then $\begin{matrix} K_{V} = = \{\begin{matrix} {a} & i f m ⩾ 2, \\ \emptyset & otherwise . \end{matrix} \end{matrix}$ And as consequence of the above result, if $m ⩾ 2$ , A compact subset of Ω $\begin{matrix} V (x) = dist {(x; A)}^{- m} then K_{V} = A . \end{matrix}$
Corollary 7.2 (of Theorem 1 and Theorem 5).

Under the same assumptions as for Theorem 5 and Theorem 1 , with $K = K_{V}$ , then for $Φ \in C_{0}^{2} (\overline{Ω})$ the sequence ${(Φ_{j})}_{j}$ given in the proof of Theorem 5 say $\begin{matrix} Φ_{j} = (1 - φ_{j}) H (\frac{2}{μ_{j}} ρ) Φ \end{matrix}$ satisfies: For all open set $Ω_{V, 0}$ relatively compact in $Ω_{V}$ one has: $\begin{array}{l} 1 . \underset{{\overline{Ω}}_{V, 0}}{Max} | Φ_{j} (x) - Φ (x) | \to_{j \to + \infty}^{} 0 \\ 2 . {| \nabla (Φ_{j} - Φ) |}_{L^{n, 1} (Ω_{V, 0})} \to_{j \to + \infty}^{} 0 . \end{array}$

Proof.
Let $Ω_{V, 0} \subset \subset Ω_{V}$ . Then according to Theorem 1 $\begin{array}{l} (50) & \underset{{\overline{Ω}}_{V, 0}}{Max} | φ_{j} (x) | \to_{j \to + \infty}^{} 0 \\ (51) & ‖ \nabla φ_{j} ‖_{L^{n, 1} (Ω_{V, 0})} \to_{j \to + \infty}^{} 0 . \end{array}$ On the other hand for $j ⩾ j_{0}$ , we have $Ω_{V, 0} \subset {ρ > μ_{j}}$ . Therefore we have $\begin{array}{l} (52) & \underset{{\overline{Ω}}_{V, 0}}{Max} | Φ_{j} (x) - Φ (x) | ⩽ ‖ Φ ‖_{\infty} \underset{{\overline{Ω}}_{V, 0}}{Max} | φ_{j} (x) | \\ (53) & {‖ \nabla (Φ_{j} - Φ) ‖}_{L^{n, 1} ({\overline{Ω}}_{V, 0})} ⩽ ‖ \nabla Φ ‖_{\infty} \underset{{\overline{Ω}}_{V, 0}}{Max} | φ_{j} (x) | + ‖ Φ ‖_{\infty} ‖ \nabla φ_{j} ‖_{L^{n, 1} (Ω_{V, 0})} \end{array}$

Relations (50) to (53) give the result. □

As in [5], we may add a transport term $U \cdot \nabla φ$ in the above equation (46).
Lemma 5.2.
Let V a nonnegative potential K be a compact in Ω with ${Cap}_{V, \infty} (K) = 0$ .

Consider $U \in L^{p, 1} {(Ω)}^{n}$ , $p > n$ , $w \in L^{1} (V) \cap L^{q} (δ^{- 1})$ , $q < p^{'}$ . Assume that $w \in L^{\frac{n}{n - 2}, \infty} (Ω)$ if $n ⩾ 3$ and $w \in L_{exp} (Ω)$ if $n = 2$ .

Then, for all $Φ \in C_{0}^{2} (\overline{Ω})$ , the sequence given in Theorem 5 , $Φ_{j} = (1 - φ_{j}) H (\frac{2}{μ_{j}} ρ) Φ$ satisfies $\begin{array}{l} 1 . lim_{j \to + \infty} \int_{Ω} w Δ Φ_{j} d x = \int_{Ω} w Δ Φ d x, \\ 2 . lim_{j \to + \infty} \int_{Ω} w Φ_{j} V d x = \int_{Ω} Φ w V d x, \\ 3 . lim_{j \to + \infty} \int_{Ω} w U \cdot \nabla Φ_{j} d x = \int_{Ω} w U \cdot \nabla Φ d x . \end{array}$
Proof.
The two first statements are the consequence of the fact $w \in L^{1} (V) \cap L_{0}^{1} (Ω)$ , $(L^{1} (δ^{- 1}) \subset L_{0}^{1} (Ω))$ and the fact that $Φ_{j}$ converges weakly-strongly to Φ in the sense of the potential V. Moreover, we have a constant $c_{H Φ} > 0$ $\begin{matrix} (54) & \int_{Ω} | w U \cdot (\nabla Φ_{j} - \nabla Φ) | d x ⩽ c_{H_{Φ}} [{‖ \frac{\nabla φ_{j}}{\sqrt{V}} ‖}_{\infty} \int_{Ω} | w | | U | \sqrt{V} d x + \int_{Ω} | φ_{j} | | w | | U | d x] . \end{matrix}$ By Hölder, $w | U |$ and $| w | | U | \sqrt{V}$ are in $L^{1} (Ω)$ since $\begin{matrix} \int_{Ω} | w | | U | d x ⩽ {‖ \frac{w}{δ} ‖}_{L^{q}} ‖ U ‖_{L^{q^{'}}} < + \infty, q < n^{'}, \frac{1}{q} + \frac{1}{q^{'}} = 1 \end{matrix}$ and $\begin{matrix} \int_{Ω} | w | | U | \sqrt{V} d x ⩽ c ‖ w ‖_{L^{1} (V)}^{\frac{1}{2}} ‖ w ‖_{L^{\frac{n}{n - 2}, \infty}}^{\frac{1}{2}} \cdot ‖ U ‖_{L^{n, 1}} if n ⩾ 3 . \end{matrix}$ Idem for $n = 2$ . Therefore, relation (54) leads to statement 3. knowing $\begin{matrix} lim_{j \to + \infty} φ_{j} (x) = 0, 0 ⩽ φ_{j} ⩽ 1 and {‖ \frac{\nabla φ_{j}}{\sqrt{V}} ‖}_{\infty} \to_{j \to + \infty}^{} 0 . \end{matrix}$ □
Theorem 8.
Under the same assumption as for Lemma 5.2 , if furthermore w satisfies $\begin{matrix} (55) & \int_{Ω} w (- Δ Φ - U \cdot \nabla Φ + V Φ) d x = 0 \forall Φ \in C_{c}^{2} (Ω ∖ K) \end{matrix}$ then, ( 55 ) holds for all $Φ \in C_{0}^{2} (\overline{Ω})$ , and if $\partial Ω \in C^{1, 1}$ and $div (\vec{U}) = 0$ in $D^{'} (Ω)$ with $\vec{U} \cdot \vec{ν} = 0$ on $\partial Ω$ (ν exterior normal to $\partial Ω$ ) then $\begin{matrix} w \equiv 0 . \end{matrix}$
Proof.
If $Φ \in C_{0}^{2} (\overline{Ω})$ we have a sequence $Φ_{j}$ in $C_{c}^{2} (Ω ∖ K)$ such that, $Φ_{j}$ satisfies the conclusion of Lemma 5.2. Thus, we have (55) with $Φ \in C_{0}^{2} (\overline{Ω})$ as test function. To prove that $w \equiv 0$ we need to employ the following variant of Kato’s inequality (see [5]). Theorem 9 (Variant of Kato’s inequality).

Let $\overline{u}$ be in $W_{loc}^{1, 1} (Ω) \cap L^{n^{'}, \infty} (Ω)$ with $\frac{\overline{u}}{δ} \in L^{1} (Ω)$ and $\vec{U} \in L^{n, 1} {(Ω)}^{n}$ with $div (\vec{U})$ in $D^{'} (Ω)$ , $\vec{U} \cdot \vec{ν} = 0$ on $\partial Ω$ .

Assume that $L \overline{u} = - Δ \overline{u} + div (\vec{U} \overline{u}) \in L^{1} (Ω; δ)$ . Then for all $ϕ \in C_{0}^{2} (\overline{Ω})$ , $ϕ ⩾ 0$ one has $\begin{array}{l} 1 . \int_{Ω} {\overline{u}}_{+} L^{*} ϕ d x ⩽ \int_{Ω} ϕ {sign}_{+} (\overline{u}) L \overline{u} d x \\ 2 . \int_{Ω} | \overline{u} | L^{*} ϕ d x ⩽ \int_{Ω} ϕ sign (μ) L \overline{u} d x, \end{array}$ where $L^{*} ϕ = - Δ ϕ - \vec{u} \cdot \nabla ϕ = - Δ - div (\vec{U} ϕ)$ .

According to equation (55), $L w = - V w \in L^{1} (Ω)$ . Thus the above Kato’s type inequality holds and $\begin{matrix} \forall ϕ \in C_{0}^{2} (\overline{Ω}), ϕ ⩾ 0 : \int_{Ω} | w | L^{*} ϕ ⩽ - \int_{Ω} ϕ | w | V d x ⩽ 0 . \end{matrix}$ Thus one has $\begin{matrix} \int_{Ω} | w | L^{*} ϕ = 0 \forall ϕ \in C_{0}^{2} (\overline{Ω}) . \end{matrix}$

By density result the same equation holds $\begin{matrix} \forall ϕ \in H_{0}^{1} (Ω) \cap W^{2} L^{n, 1} (Ω), ϕ ⩾ 0 . \end{matrix}$ Resolving $L^{*} ϕ = 1$ we derive that $w \equiv 0$ . □

Next, we want to discuss some existence problem related to equation (55).

We always assume that $U \in L^{p, 1} {(Ω)}^{n}$ , $p > n$ , $div (U) = 0$ in $D^{'} (Ω)$ , $U \cdot ν = 0$ on $\partial Ω$ and $\partial Ω \in C^{1, 1}$ .

Theorem 10.

Let f be a bounded Radon measure in Ω. Assume that ${Cap}_{V, \infty} (K_{V}) = 0$ .

If $| f | (K_{V}) = 0$ (f does not charge the compact set $K_{V}$ ) then there exists an unique solution $u \in L^{1} (V) \cap L^{1} (Ω; δ^{- 1})$ such that $\begin{matrix} (56) & \int_{O} u (- Δ φ - U \cdot \nabla φ + V φ) d x = \int_{Ω} φ d f \forall φ \in C_{0}^{2} (\overline{Ω}) . \end{matrix}$

Proof.

The uniqueness is a consequence of Theorem 8.

For the existence, we first notice that the problem is linear, we may assume that $f ⩾ 0$ . We shall set as usual $\begin{array}{l} M^{1} (Ω) = {f : bounded Radon measure on Ω}, \\ M^{1} (Ω) = C_{0} {(\overline{Ω})}^{'}, C_{0} (\overline{Ω}) = {φ : \overline{Ω} \to R continuous, φ = 0 on \partial Ω} \end{array}$ Let us introduce $V_{j} = min (j; V)$ we have proved the following result in [4,5]. □

Lemma 5.3.

There exists $u_{j} ⩾ 0$ , $u_{j} \in W_{0}^{1} L^{n^{'}, \infty} (Ω)$ such that

$\forall φ \in H_{0}^{1} (Ω) \cap W^{2} L^{n, 1} (Ω)$ $\begin{matrix} (57) & \int_{Ω} u_{j} [- Δ φ - U \cdot \nabla φ + V_{j} φ] d x = \int_{Ω} φ d f . \end{matrix}$

There exists a constant $c_{0}$ independent of j such that $\begin{matrix} (58) & ‖ u_{j} ‖_{W_{0}^{1} L^{n^{'}, \infty} (Ω)} + \int_{Ω} V_{j} u_{j} d x ⩽ c_{0} ‖ f ‖_{M^{1} (Ω)} . \end{matrix}$

In particular, there exist a function $u ⩾ 0$ and a subsequence $u_{j}$ such that $\begin{array}{l} (a) u_{j} \to_{j \to + \infty}^{} u (x) a.e. in Ω, strongly in L^{1} (Ω) and weakly in W_{0}^{1} L^{n^{'}, \infty} (Ω) . \\ (59) & (b) ‖ u ‖_{W_{0}^{1} L^{n^{'}, \infty} (Ω)} + \int_{Ω} V u d x ⩽ c_{0} ‖ f ‖_{M^{1} (Ω)} . \end{array}$

Proof of Lemma 5.3.

Since $f \in M^{1} (Ω)$ , there is a sequence $f_{k} \in L_{+}^{\infty} (Ω)$ such that $\begin{matrix} ‖ f_{k} ‖_{L^{1} (Ω)} ⩽ ‖ f ‖_{M^{1} (Ω)} and f_{k} converges to f weakly in C_{c} {(Ω)}^{'} \end{matrix}$ (i.e. $\forall φ \in C_{c} (Ω)$ $⟨ f_{k}, φ ⟩ \to ⟨ f, φ ⟩$ ).

According to [4,5], one has a function $u_{j k} \in W_{0}^{1} L^{n^{'}, \infty} (Ω)$ satisfying, $\forall φ \in C_{0}^{2} (\overline{Ω})$ $\begin{matrix} (60) & \int_{Ω} u_{j k} [- Δ φ - U \cdot \nabla φ + V_{j} φ] d x = \int_{Ω} f_{k} φ d x \end{matrix}$ and $\begin{matrix} (61) & ‖ u_{j k} ‖_{W^{1} L^{n^{'}, \infty} (Ω)} + \int_{Ω} V_{j} u_{j k} d x ⩽ c_{0} ‖ f_{k} ‖_{L^{1} (Ω)} ⩽ c_{0} ‖ f ‖_{M^{1} (Ω)} \end{matrix}$ where $c_{0}$ is independent of j and k (in fact $c_{0}$ depends on Ω and $‖ U ‖_{L^{n, 1} (Ω)}$ ). More $u_{j k} ⩾ 0$ . Thus we have a subsequence still denoted ${(u_{j k})}_{k}$ and a function $u_{j} \in W_{0}^{1} L^{n^{'}, \infty} (Ω)$ , $u_{j} ⩾ 0$ such $u_{j k} ⇀_{k \to + \infty} u_{j}$ weakly in $W_{0}^{1} L^{n^{'}, \infty} (Ω)$ , strongly in $L^{1} (Ω)$ and almost everywhere in Ω.

Thus, we can pass easily to the limit in relations (60) and (61) to derive the part 1.) and 2.) of Lemma 5.3. By the same reason as above, we have a subsequence still denoted $u_{j}$ and a function $u ⩾ 0$ such that $u_{j} ⇀ u$ weakly in $W_{0}^{1} L^{n^{'}, \infty} (Ω)$ strongly in $L^{1} (Ω)$ , almost everywhere in Ω. From relation (61) using among other Fatou’s lemma, we have relation (59). □

Lemma 5.4.

Let $φ \in W_{0}^{1} L^{n, 1} (Ω)$ with $support (φ) \cap K_{V} = \emptyset$ .

Then $\begin{matrix} lim_{j \to + \infty} \int_{Ω} | u_{j} V_{j} - u V | | φ | d x = 0 . \end{matrix}$

Proof.

Let φ be in $W_{0}^{1} L^{n, 1} (Ω)$ with $support (φ) \cap K_{V} = \emptyset$ .

Thus $V φ \in L^{\frac{n}{2}, 1} (Ω)$ and $support (V φ) \subset \subset Ω ∖ K_{V} = Ω_{V}$ , since $support (φ) \subset {x : dist (x; K) > η}$ for some $η > 0$ . We have: $\begin{array}{rcl} (62) & \begin{array}{l} ‖ u_{j} - u ‖_{L_{exp} (Ω)} ⩽ c_{N} ‖ u_{j} - u ‖_{W_{0}^{1} L^{2, \infty} (Ω)} if n = 2 \\ ‖ u_{j} - u ‖_{L^{\frac{n}{n - 2}, \infty} (Ω)} ⩽ c_{N} ‖ u_{j} - u ‖_{W_{0}^{1} L^{n^{'}, \infty} (Ω)} if n ⩾ 3 . \end{array} \end{array}$ Therefore, applying Hölder’s inequality, we have a constant $c_{6} > 0$ (independent of $u_{j}$ , u, $V_{j}$ ) such that for any measurable subset $E \subset Ω$ $\begin{matrix} (63) & \int_{E} V | φ | | u_{j} - u | d x ⩽ c_{6} ‖ V φ ‖_{L^{\frac{n}{2}, 1} (E)} \to_{| E | \to 0}^{} 0 . \end{matrix}$ Therefore, using the Egoroff’s theorem or Vitali’s theorem one has $\begin{matrix} (64) & lim_{j \to \infty} \int_{Ω} V | φ | | u_{j} - u | d x = 0 . \end{matrix}$ Since we have $\begin{matrix} (65) & lim_{j \to \infty} \int_{Ω} | u | | V_{j} - V | | φ | d x = 0 . \end{matrix}$ Finally we have $\begin{matrix} (66) & lim_{j \to \infty} \int_{Ω} | u_{j} V_{j} - u V | | φ | d x = 0 . \end{matrix}$ □

Lemma 5.5.

The function u found in the preceding Lemma 5.3 , satisfies, $\forall φ \in C_{c}^{2} (Ω_{V})$ $\begin{matrix} \int_{Ω} u (- Δ φ - U \cdot \nabla φ + V φ) d x = \int_{Ω} φ d f . \end{matrix}$

Proof.

Let φ be in $C_{c}^{2} (Ω_{V})$ then, $\begin{matrix} (67) & \int_{Ω} u_{j} [- Δ φ - U \cdot \nabla φ + V_{j} φ] d x = \int_{Ω} φ d f \end{matrix}$ and $\begin{matrix} lim_{j \to + \infty} \int_{Ω} | u_{j} V_{j} - u V | | φ | d x = 0 (since support (φ) \cap K_{V} = \emptyset) . \end{matrix}$ Thus we may pass to the limit in relation (67). □

Lemma 5.6.

If ${Cap}_{V, \infty} (K_{V}) = 0$ and $| f | (K_{V}) = 0$ then, $\begin{matrix} the function u given in Lemma 5.5 satisfies relation (56) \end{matrix}$ and $\begin{matrix} u \in L^{1} (V) \cap W_{0}^{1} L^{n^{'}, \infty} (Ω) \subset L^{1} (V) \cap L^{q} (Ω; \frac{1}{δ}), q < n^{'} . \end{matrix}$

Proof.

Let Φ be in $C_{0}^{2} (\overline{Ω})$ . From our assumption we have a sequence $Φ_{j} \in C_{c}^{2} (Ω_{V})$ such that:

$Φ_{j}$ converges weakly to Φ in the sense of potential V,

$Φ_{j} (x) \to_{j \to + \infty} Φ (x)$ for all $x \in Ω - K_{V} = Ω_{V}$ (see Corollary 7.2 of Theorem 5 and Theorem 1), $Φ_{j} = 0$ in the neighborhood of $K_{V}$

Since

\begin{matrix} (68) & \int_{Ω} u (- Δ Φ_{j} - U \cdot \nabla φ_{j} + V Φ_{j}) d x = \int_{Ω} Φ_{j} d f . \end{matrix}

We pass to the limit since

u \in L^{1} (V) \cap W_{0}^{1} L^{n^{'}, \infty} Ω \subset L^{1} (V) \cap L_{0}^{1} (Ω)

in the first integral and in the second integral using Lebesgue dominated theorem to derive

\begin{array}{l} (69) & lim_{j \to \infty} \int_{Ω} u (- Δ Φ_{j} + V Φ_{j}) d x = \int_{Ω} u (- Δ Φ + V Φ) d x \\ (70) & lim_{j \to \infty} \int_{Ω} Φ_{j} d f = \int_{Ω ∖ K_{V}} Φ d f . \end{array}

Applying Lemma 5.3, we have

\begin{matrix} (71) & lim_{j \to \infty} \int_{Ω} u U \cdot \nabla Φ_{j} d x = \int_{Ω} u U \cdot \nabla Φ d x . \end{matrix}

But

| f | (K_{V}) = 0

so we have

\begin{matrix} (72) & \int_{Ω ∖ K_{V}} Φ d f = \int_{Ω} Φ d f . \end{matrix}

From relations (68) to(72), we derive

\begin{matrix} (73) & \int_{Ω} u (- Δ Φ - U \cdot \nabla Φ + V Φ) d x = \int_{Ω} Φ d f . \end{matrix}

□

For the converse, we will first prove

Theorem 11.

Assume that ${Cap}_{V, \infty} (K_{V}) = 0$ , $f = μ_{a}$ , the Dirac measure at $a \in Ω$ .

If $a \in K_{V}$ then there is no solution $u \in L^{1} (V) \cap L_{0}^{1} (Ω)$ of $\begin{matrix} (74) & \int_{Ω} u [- Δ φ - U \cdot \nabla φ + V φ] d x = φ (a) \forall φ \in C_{0}^{2} (\overline{Ω}) . \end{matrix}$

Proof.

If there was a solution, then, $\forall φ \in C_{c}^{2} (Ω ∖ K_{V})$ we have $\begin{matrix} \int_{Ω} u [- Δ φ - U \cdot \nabla φ + V φ] d x = 0 . \end{matrix}$ But ${Cap}_{V, \infty} (K_{V}) = 0$ thus the same equation holds for all $φ \in C_{0}^{2} (\overline{Ω})$ which implies that $\forall φ \in C_{0}^{2} (\overline{Ω})$ , $φ (a) = 0$ . This is impossible. □

One can generalize Theorem 11 as follow

Theorem 12.

Assume that ${Cap}_{V, \infty} (K_{V}) = 0$ . Let f be a bounded Radon measure such that $G = support (f) \cap K_{V}$ is an isolate subset of $support (f)$ , i.e. there exists an open set ω such that $ω \cap support (f) = G$ . $\begin{matrix} (75) & if | f | (K_{V}) > 0 then there is no solution of (73) . \end{matrix}$

Proof.

If $| f | (K_{V}) > 0$ , then G is an isolate subset of support of f, therefore, we can consider $θ \in C_{c}^{\infty} (Ω)$ such that $θ = 1$ on G, $support θ \subset ω$ .

We write $f = θ f + (1 - θ) f$ so that measure $f_{1} = (1 - θ) f$ does not charge $K_{V}$ . By the preceding result, we have $u_{1} \in L^{1} (V) \cap L_{0}^{1} (Ω)$ such that $\begin{matrix} \int_{Ω} u_{1} [- Δ φ - U \cdot \nabla φ + V φ] d x = \int_{Ω} φ d f_{1} \forall φ \in C_{0}^{2} (\overline{Ω}) . \end{matrix}$ Assume that we have a solution u of (73) so, $w = u - u_{1}$ is a solution of $\begin{matrix} \int_{Ω} w [- Δ φ - U \cdot \nabla φ + V φ] d x = ⟨ θ f, φ ⟩, \forall φ \in C_{0}^{2} (\overline{Ω}) . \end{matrix}$ In particular $\begin{matrix} \int_{Ω} w [- Δ φ - U \cdot \nabla φ + V φ] = 0 \forall φ \in C_{c}^{2} (Ω ∖ K_{V}) . \end{matrix}$ Applying Theorem 8, we deduce that $w = 0$ say $u = u_{1}$ which mean $f = (1 - θ) f$ : $θ f \equiv 0$ this is a contradiction with the fact that $| f | (K_{V}) > 0$ . □

Theorem 13.

Assume that ${Cap}_{V, \infty} (K_{V}) = 0$ , $f \in M^{1} (Ω)$ such that $G = support (f) \cap K_{V}$ is an isolate subset of $support (f)$ .

Then one has a solution $u \in L^{1} (V) \cap L_{0}^{1} (Ω)$ of $\begin{matrix} \int_{Ω} u [- Δ φ - U \cdot \nabla φ + V φ] = \int_{Ω} φ d f \forall φ \in C_{0}^{2} (\overline{Ω}) if and only if | f | (K_{V}) = 0 . \end{matrix}$

After submitting this work, we have received the paper [11] where a similar result as for this last theorem is given but only for solution in $W_{0}^{1, 1} (Ω) \cap L^{1} (V)$ which is strictly included in $L_{0}^{1} (Ω) \cap L^{1} (V)$ .

More, our proofs are totally different.

This paper is also published on preprint server arXiv:1812.04061v1 [math.AP] 10 Dec 2018.

Footnotes

Acknowledgements

This work was initiated partly in December 18th, 2017 when the author attended to the conference “Nonlinear Partial Equation and Mathematical Analysis”.

He would like to thank all the participants and the organizers for their warm hospitality and invitation. A special thanks to Prof. Díaz Ildefonso whose friendship is a constant encouragement and an inspiration for him.

He wants also to thank the anonymous referees for reading carefully this manuscript.

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Potential-capacity and some applications

Abstract

Keywords

1. Introduction

2. Notations definitions – primary definitions and results

Definition 1 (of the distribution function and monotone rearrangement).

Property 1 (of ( V , ∞ ) -capacity).

Theorem 2 (Comparison near a compact).

Proof. Let φ ∈ C 0 2 ( Ω ‾ ) then φ j H ( j ρ ( x ) ) is in C c 2 ( Ω ∖ A ) and the above arguments imply the statement. □ 5. Applications of the potential-capacity and the approximation of C 0 2 ( Ω ‾ )

Corollary 7.1 (of Theorem 7 and Lemma 5.1).

Footnotes

Acknowledgements

References

Property 1 (of $(V, \infty)$ -capacity).

Proof.
Let $φ \in C_{0}^{2} (\overline{Ω})$ then $φ_{j} H (j ρ (x))$ is in $C_{c}^{2} (Ω ∖ A)$ and the above arguments imply the statement. □

5. Applications of the potential-capacity and the approximation of $C_{0}^{2} (\overline{Ω})$