Arnaud Dumont | Searcharxiv

arXiv2025

Tangential approach in the Dirichlet problem for elliptic equations

It is well-known that solvability of the $\mathrm{L}^{p}$-Dirichlet problem for elliptic equations $Lu:=-\mathrm{div}(A\nabla u)=0$ with real-valued, bounded and measurable coefficients $A$ on Lipschitz domains $Ω\subset\mathbb{R}^{1+n}$ is characterised by a quantitative absolute continuity of the associated $L$-harmonic measure. We prove that this local $A_{\infty}$ property is sufficient to guarantee that the nontangential convergence afforded to $\mathrm{L}^{p}$ boundary data actually improves to a certain \emph{tangential} convergence when the data has additional (Sobolev) regularity. Moreover, we obtain sharp estimates on the Hausdorff dimension of the set on which such convergence can fail. This extends results obtained by Dorronsoro, Nagel, Rudin, Shapiro and Stein for classical harmonic functions in the upper half-space.

math.AP↗

arXiv2025

Boundary value problems and Hardy spaces for singular Schrödinger equations with block structure

We obtain Riesz transform bounds and characterise operator-adapted Hardy spaces to solve boundary value problems for singular Schrödinger equations $-\mathrm{div}(A\nabla u)+aVu=0$ in the upper half-space $\mathbb{R}^{1+n}_{+}$ with boundary dimension $n\geq 3$. The coefficients $(A,a,V)$ are assumed to be independent of the transversal direction to the boundary, and consist of a complex-elliptic pair $(A,a)$ that is bounded and measurable with a certain block structure, and a non-negative singular potential $V$ in the reverse Hölder class $\mathrm{RH}^{q}(\mathbb{R}^{n})$ for $q\geq \max\{\frac{n}{2},2\}$. This block structure is significant because it allows for coefficients that are not symmetric but for which $\mathrm{L}^{2}(\mathbb{R}^{n})$-solvability persists due to recently obtained Kato square root type estimates. We find extrapolation intervals for exponents $p$ around $2$ on which the Dirichlet problem is well-posed for boundary data in $\mathrm{L}^{p}(\mathbb{R}^{n})$, and the associated Regularity problem is well-posed for boundary data in Sobolev spaces $\dot{\mathcal{V}}^{1,p}(\mathbb{R}^{n})$ that are adapted to the potential $V$, when $p>1$. The well-posedness of these Dirichlet problems and related estimates then allow us to solve the corresponding Neumann problem with boundary data in $\mathrm{L}^{p}$. The results permit boundary data in the Dziubanski--Zienkiewicz Hardy space $\mathrm{H}^{1}_{V}(\mathbb{R}^{n})$ and adapted Hardy--Sobolev spaces $\dot{\mathrm{H}}^{1,p}_{V}(\mathbb{R}^{n})$ when $p\leq 1$. We also obtain comparability of square functions and nontangential maximal functions for the solutions with their boundary data.

math.AP↗