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This is in fact quite elementary: if $\Omega$ is smooth, then there is $r > 0$ such that at every boundary point $y$ of $\Omega$ there is a tangent ball $B(z, r)$ of radius $r$ which is contained in $ …

Dini continuity may be what you are looking for: if $f = \Delta u$ is Dini continuous, that is, $$\int_0^1 \frac{\omega_f(t)}{t} \, dt < \infty,$$ then $u$ is $C^2$. This is a rather old result, but I …

$\newcommand{\R}{\mathbb R}$Sorry for being too sketchy in the following answer, time permitting, I'll try to expand. Step 0. Some more-or-less classical potential theory. Let $D$ be an open set in $ …

(For an actual answer, see the edit below.) Let $\phi$ be smooth near zero and non-negative. Suppose that the Taylor expansion of $\phi$ at zero is non-trivial, and let $P(x)$ be the leading term. The …

(This is an extended comment, I suppose). What do we assume about $L$? In complete generality this cannot be true, as shown by the example that follows. Even if we assume, as in the paper that you li …

Martin kernel and Martin representation is what you are after. Positive solutions are: $$ u(x) = \int_{\partial B_r} \frac{(1 - |x|^2)^s}{|x - y|^N} \, \mu(dy) $$ for any positive measure $\mu$. Signe …

We claim the following: a solution $u$ exists if and only if $f$ is orthogonal to the Poisson kernel with pole at every $x \in \partial \Omega$. Suppose that $u$ is a solution. Since $u = 0$ on the …

Edit: The result is fine: Hopf's lemma was proved in G. Giraud, Problèmes de valeurs à la frontière relatifs à certaines donn ás discontinues, Bull. de la Soc. Math. de France, 61 (1933), 1–54 Below …

Yes, this is possible. An explicit example is $$u(x, y, z) = 1 - I_0\left(\sqrt{x^2 + y^2}\right) \, \cos z$$ when $L = \pi$, and $u\big(\frac{\pi x}{L}, \frac{\pi x}{L}, \frac{\pi x}{L}\big)$ for a g …

Write $w(x) = u(x) + c (1 - |x|^2)$ with $u \geqslant 0$ harmonic in $B_1(0)$, and apply standard Harnack's inequality: $$w(0) = u(0) + c \leqslant C^{-1} u(y) + \tfrac{4}{3} c (1 - |y|^2) \le \max(C^ …

This is an extended comment, not an answer. Suppose that $a_0$ and $b_0$ satisfy the conditions given in the statement of the problem and $u := b_0-a_0$ is not identically zero. The function $u$ is o …

Just an extended comment. Theorem 4.1 in The Cauchy process and the Steklov problem by Rodrigo Bañuelos and Tadeusz Kulczycki (JFA 2004, DOI:10.1016/j.jfa.2004.02.005) shows that the eigenfunctions $ …

OK, here are some additional details to what I wrote in my comments. Due to scaling, we can choose $r = 1$. Let $D = B_1 \setminus B_k$, and let $P_D(x, y)$ be the Poisson kernel of $D$. Thus, $$ w(x) …

Here are some additional details to what I wrote under the previous question. Our key tool is the following result. Bochner's relation: Let $V(x)$ be a solid harmonic polynomial of degree $\ell$: a ho …

I believe only partial results are available, and the decay is typically slower than exponential. Here is what I am aware of when $V(x) \to \infty$ as $|x| \to \infty$ (in this case the spectrum is pu …