arXiv subjects

Nicolas Wilmet

Publications and source records attributed to Nicolas Wilmet.

4 recordsLinked to original sources

arXiv2025

A representation formula for the distributional normal derivative

We prove an integral representation formula for the distributional normal derivative of solutions of $$ \left\{ \begin{aligned} - Δu + V u &= μ&& \text{in $Ω$,}\\ u &= 0 && \text{on $\partialΩ$,} \end{aligned} \right. $$ where $V \in L_{\mathrm{loc}}^1(Ω)$ is a nonnegative function and $μ$ is a finite Borel measure on $Ω$. As an application, we show that the Hopf lemma holds almost everywhere on $\partialΩ$ when $V$ is a nonnegative Hopf potential.

math.AP↗

arXiv2020

The Hopf lemma for the Schrödinger operator

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator $- Δ+ V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(Ω)$. More precisely, if $u \in W_0^{1, 2}(Ω) \cap L^2(Ω; V \mathrm{d}x)$ satisfies $- Δu + V u = f$ on $Ω$ for some nonnegative datum $f \in L^\infty(Ω)$, $f \not\equiv 0$, then we show that at every point $a \in \partialΩ$ where the classical normal derivative $\partial u(a) / \partial n$ exists and satisfies the Poisson representation formula, one has $\partial u(a) / \partial n > 0$ if and only if the boundary value problem $$ \begin{cases} \begin{aligned} - Δv + V v &= 0 && \text{in $Ω$,} \\ v &= ν&& \text{on $\partialΩ$,} \end{aligned} \end{cases} $$ involving the Dirac measure $ν= δ_a$ has a solution. More generally, we characterize the nonnegative finite Borel measures $ν$ on $\partialΩ$ for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

math.AP↗

arXiv2017

Optimal control of nonlinear elliptic problems with sparsity

We study the minimization of the cost functional \[ F(μ) = \lVert u - u_d \rVert_{L^p(Ω)} + α\lVert μ\rVert_{\mathcal{M}(Ω)}, \] where the controls $μ$ are taken in the space of finite Borel measures and $u \in W_0^{1, 1}(Ω)$ satisfies the equation $- Δu + g(u) = μ$ in the sense of distributions in $Ω$ for a given nondecreasing continuous function $g : \mathbb{R} \to \mathbb{R}$ such that $g(0) = 0$. We prove that $F$ has a minimizer for every desired state $u_d \in L^1(Ω)$ and every control parameter $α> 0$. We then show that when $u_d$ is nonnegative or bounded, every minimizer of $F$ has the same property.

math.AP↗

arXiv2017

Schroedinger operators involving singular potentials and measure data

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$ \left\{ \begin{alignedat}{2} -Δu + Vu & = μ&& \quad \text{in } Ω,\\ u & = 0 && \quad \text{on } \partial Ω. \end{alignedat} \right. $$ We characterize the finite measures $μ$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(Ω)$ with $1 \le p \le N/2$. The full answer can be expressed in terms of the $W^{2,p}$ capacity for $p > 1$, and the $W^{1,2}$ (or Newtonian) capacity for $p = 1$. We then prove the existence of a solution of the problem above when $V$ belongs to the real Hardy space $H^1(Ω)$ and $μ$ is diffuse with respect to the $W^{2,1}$ capacity.

math.AP↗