[Paper Review] Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces

Marcelo F. de Almeida, Lidiane S. M. Lima|arXiv (Cornell University)|Jul 11, 2018

Nonlinear Partial Differential Equations18 references3 citations

TL;DR

This paper establishes the existence of positive solutions to a nonlinear Neumann boundary problem for harmonic functions in higher-dimensional half-spaces, using a novel framework based on weak-Morrey spaces. By introducing a Banach space of functions with controlled integrability and applying contraction mapping in this space, the authors prove existence for singular data $ f $ and potential $ V $ in weak-Morrey spaces when $ n/(n-1) < \rho < \infty $, recovering and extending known results for the half-Laplacian problem.

ABSTRACT

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -Δu=0 &amp; ext{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial ν}} = V(x)u+b \vert u\vert^{ρ-1}u+f \; &amp; ext{on} \;\;\partial\mathbb{R}^{n+1}_+\;\;\;\;\;\;\;\;\,\, \end{cases} \end{align*} %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data $f$ and potential $V$ on boundary $\partial\mathbb{R}^{n+1}_+$ of half-space $ \mathbb{R}^{n+1}_{+}=\{(x,t)\in\mathbb{R}^{n+1}\,\vert\, t&gt;0\}$ for $n\geq 2$. More precisely, inspired at \cite{deAlmeida1} and \cite{Quittner} we introduce a new functional space based in weak-Morrey spaces and we shown existence of positive solutions $u$ to the above problem when inhomogeneous term $f\in ext{weak-}\mathcal{M}_{p}^{n{(ρ-1)}/ρ}(\mathbb{R}^{n})$ and potential $V\in ext{week-}\mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ are sufficiently small in the natural $n/(n-1)

Motivation & Objective

• To establish well-posedness of a nonlinear boundary value problem for harmonic functions in $ \mathbb{R}^{n+1}_+ $ with rough data.
• To extend solvability results to inhomogeneous terms $ f $ and potentials $ V $ in weak-Morrey spaces, which are larger than classical Lebesgue or Morrey spaces.
• To recover the range $ (n+1)/(n-1) \leq \rho < \infty $ for the nonlinearity exponent $ \rho $, improving upon previous results.
• To show that solutions are locally Hölder continuous via Campanato’s lemma, even when $ f $ and $ V $ are rough.
• To connect the local problem to the nonlocal half-Laplacian equation via the Dirichlet-to-Neumann map.

Proposed method

• Introduce a new Banach space $ X^{1,\infty,\infty}_{rq}(\mathbb{R}^{n+1}_+) $ to handle integrability of traces and gradients in irregular domains.
• Define the solution operator $ \Psi(u) = \mathbf{S}f + \mathcal{T}_V(u) + \mathcal{B}(u) $, combining single layer potential, potential term, and nonlinear term.
• Apply the contraction mapping principle in $ X^{1,\infty,\infty}_{rq}(\mathbb{R}^{n+1}_+) $, showing $ \Psi $ is a contraction when $ \|V\|_{\mathcal{M}^{n}_{\ell,\infty}} < 1 $ and $ \|f\|_{\mathcal{M}^{\omega}_{p,\infty}} $ is small.
• Use estimates involving weak-Morrey norms and fractional maximal functions to control the nonlinear and potential terms.
• Leverage the relation between the half-Laplacian and the Dirichlet-to-Neumann map to link the local problem to nonlocal equations.
• Apply Campanato’s lemma to prove local Hölder regularity of the solution $ u \in C^{0,\alpha}_{\text{loc}}(\overline{\mathbb{R}^{n+1}_+}) $.

Experimental results

Research questions

• RQ1 Can the solvability range of the nonlinear Neumann problem for harmonic functions be extended to rougher data than previously known?
• RQ2 What function space framework allows for existence of solutions when the inhomogeneous term $ f $ and potential $ V $ are in weak-Morrey spaces?
• RQ3 Does the solution to the local harmonic problem with nonlinear Neumann data correspond to a solution of the nonlocal half-Laplacian equation?
• RQ4 How does the use of weak-Morrey spaces improve the range of admissible exponents $ \rho $ in the nonlinearity $ |u|^{\rho-1}u $?
• RQ5 Can local Hölder regularity of the solution be established under minimal integrability assumptions on $ f $ and $ V $?

Key findings

• The paper proves existence of a positive solution $ u \in X^{1,\infty,\infty}_{rq}(\mathbb{R}^{n+1}_+) $ to the nonlinear Neumann problem when $ \|f\|_{\mathcal{M}^{n(\rho-1)/\rho}_{p,\infty}(\mathbb{R}^n)} \leq \varepsilon/C $ and $ \|V\|_{\mathcal{M}^n_{\ell,\infty}(\mathbb{R}^n)} < 1 $ for sufficiently small $ \varepsilon > 0 $.
• The solution is locally Hölder continuous: $ u \in C^{0,\alpha}_{\text{loc}}(\overline{\mathbb{R}^{n+1}_+}) $, established via Campanato’s lemma.
• The method recovers the sharp range $ (n+1)/(n-1) \leq \rho < \infty $, which was previously inaccessible with classical function spaces.
• The solution framework extends to the nonlocal half-Laplacian problem $ (-\Delta)^{1/2}v = Vv + b|v|^{\rho-1}v + f $, with $ f $ and $ V $ in weaker spaces than $ L^\lambda $.
• The inclusion chain $ L^\lambda \subset \mathcal{M}^\lambda_p \subset \text{weak-}\mathcal{M}^\lambda_p $ for $ 1 < p < \lambda < \infty $ allows for rougher data than previously considered.
• The contraction mapping argument ensures uniqueness of the solution in a small ball of the Banach space $ X^{1,\infty,\infty}_{rq}(\mathbb{R}^{n+1}_+) $.