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A metric potential capacity: some qualitative properties of Schrödinger’s equations with a non negative potential

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We introduce a new notion that we call a metric potential capacity associated to a positive potential V and a weak-star topology of a suitable Banach space. We show among other things that if the metric potential capacity of a closed set F is zero and the square root of the potential is locally q-integrable for some \(q\in [1,{+\,\infty }[\), then the usual Sobolev capacity in \(W^{1,q}\) of the set F is zero. Properties of potential capacities are presented and applications to Schrödinger type equations are also made with very singular potentials including the modified Pösh–Teller potential \(V_\alpha (x)=\Big |\sin |x|\Big |^{-\alpha },\ \alpha >0\). We discuss about existence and non existence of a solution.

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Acknowledgements

The author would like to thank the anonymous referee for her/his valuable remarks.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348-SP2MI, Bat H3 Site du Futuroscope-11 Bd Marie et Pierre Curie, Téléport 2, TSA 61125, 86073, Poitiers Cedex 9, France

    Jean Michel Rakotoson

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  1. Jean Michel Rakotoson
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Correspondence to Jean Michel Rakotoson.

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This paper is dedicated to my friend Ildefonso DIAZ for his seventieth birthday.

Thank you Ilde for these 30 years of friendships.

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Appendix

Appendix

1.1 Definitions of an Assouad dimension, and Lemma of Aikawa

Let us consider a set \(E\subset \Omega ,\ r>0\) and denote by N(E, r) the minimal number of open balls of radius r centered at a point of E necessary to cover E. Then, we have

Definition 5.1

Assouad dimension of a set K [2, 22] Let \(K\subset \Omega \). The (upper) Assouad dimension of K is:

$$\begin{aligned}{} & {} \dim _A(K)\\{} & {} \quad =\inf \left\{ \lambda \ge 0\ N\Big (K\cap B(x;r)\Big )\le c_\lambda \left( \dfrac{r}{R}\right) ^{-\lambda },\ \forall \,x\in E,\ 0<r<R<diameter(K)\right\} .\end{aligned}$$

To fix an idea of such dimension, if K is a smooth sub-manifold whose Hausdorff dimension is \(\lambda \in [0,n]\), then \(\dim _A(K)=\lambda \).

The link between the Assouad dimension and the integrability of the function \(\textrm{dist}\,(y;K)\dot{=}d_K(y)\) is made via Aïkawa condition, which reads as:

Definition 5.2

Aïkawa’s condition Let K be a closed set of \(\mathrm{I\!R}^n\) and define:

$$\begin{aligned} \mathcal A(K)=\left\{ s\in \mathrm{I\!R}_+:\exists c>0,\int _{B(x;r)}\textrm{dist}\,(y;K)^{s-n}dy\le cr^s\hbox { holds if }x\in K,\ 0<r<\hbox {diameter of }K \right\} .\end{aligned}$$

Lemma 5.1

(See [21, 22]) Let K be a closed non void set. Then

  1. (1)

    \(\dim _A(K)=\inf \mathcal A(K)\le n.\)

  2. (2)

    If s is such that \(\dim _A(K)<s\) or \(s\ge n\) then \(s\in \mathcal A(K)\).

  3. (3)

    If \(s\in \mathcal A(K)\) with \(0<s<n\) then there is \(0<s'<s\) such that \(s'\in \mathcal A(K)\) in particular \(\dim _A(K)<s\).

  4. (4)

    Assume that K is a compact subset of \(\mathrm{I\!R}^n\) and let \(s\ge 0\). Then \(s\in \mathcal A(K)\) if and only if for some \(0<R\le {+\,\infty }\), there exists a constant \(c>0\) such that the inequality in \(\mathcal A(K) \) holds for \(x\in K,\ 0<r<R\).

1.2 Definition of an Hausdorff measure

Definition 5.3

(See [26, 32]) Let \(0\le s<{+\,\infty },\ 0<\delta \le {+\,\infty }\), a compact \(K\subset \mathrm{I\!R}^n\).

Then

$$\begin{aligned}\mathcal H_\delta ^s(K)=\inf \left\{ \sum _{i=0}^\ell \omega _sr_i^s,\ K\subset \bigcup _{i=0}^\ell B(x_i;r_i),\ 0<r_i\le \delta \right\} \end{aligned}$$

where \(\omega _s=\pi ^{\frac{s}{2}}\left[ \Gamma \left( \dfrac{s}{2}+1\right) \right] ^{-1}.\)

The Hausdorff measure of dimension \(s\in \mathrm{I\!R}_+\) of K is

$$\begin{aligned}\mathcal H^s(K)=\lim _{\delta \rightarrow 0}\mathcal H^s_\delta (K).\end{aligned}$$

It is shown (see for instance [26]) that \(\mathcal H^s(K)=0\) if and only if \(\mathcal H^s_\infty (K)=0\).

So the Hausdorff dimension of a compact \(K\subset \mathrm{I\!R}^n\) is

$$\begin{aligned}\dim _\mathcal H(K)=\inf \Big \{s\ge 0,\ \mathcal H^s_\infty (K)=0\Big \}.\end{aligned}$$

If \(s=n\) the \(meas(K)=\mathcal H^n(K)\).

1.3 Approximation Lemma of a distance function

We need the following Lemma whose proof is given in [20, 29, 32].

Lemma 5.2

Let \(A\subset \mathrm{I\!R}^n\) be closed and for \(x\in \mathrm{I\!R}^n\) let \(d(x)=d(x;A)\) denote the distance from x to A. Let

$$\begin{aligned}U=\Big \{x:d(x)<1\Big \}.\end{aligned}$$

Then there is a function \(\rho \in C^\infty (U-A)\) and a positive number \(M=M(n)\) such that

$$\begin{aligned}{} & {} M^{-1}d(x)\le \rho (x)\le M\,d(x),\ x\in U-A\\{} & {} |D^\alpha \rho (x)|\le c(\alpha )\,d(x)^{1-|\alpha |},\ x\in U-A,\ \ |\alpha |=\alpha _1+\cdots +\alpha _n.\end{aligned}$$

In particular, the result holds if \(A=\partial \Omega \) boundary of an open bounded set \(\Omega \) , in this case

$$\begin{aligned}\rho \in C^\infty (\Omega )\hbox { and }d(x)=\delta (x)=\textrm{dist}\,(x;\partial \Omega ).\end{aligned}$$

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Rakotoson, J.M. A metric potential capacity: some qualitative properties of Schrödinger’s equations with a non negative potential. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 112 (2022). https://doi.org/10.1007/s13398-022-01254-0

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