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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 …

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 …

In dimensions 3 and higher, and without any constraints on $\sigma$, one can apparently obtain any symmetric matrix $A = (a_{ij})$ such that $a_{ij} < 0$ when $i \ne j$ and $a_{ii} = -\sum_{j \ne i} a …

Edit: I misread the question. New answer: The question asks whether the Poisson kernel $P_\Omega(0, \cdot)$ is constant only when the domain $\Omega$ is a ball centred at $0$. This is indeed true: let …

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 …

$\newcommand{\R}{\mathbb R}$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 …

Yes, they are equivalent. Up to a constant missing and a sign error in the displayed formula, it should read: $$ (-\Delta_g)^s f(x) = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_g} f(x) - f(x)) t^ …

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 …

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 …

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) …

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 …

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 $ …

I you are not really interested about viscosity solutions, but just the "philosophical reason" why the solution of the problem with inhomogeneous exterior condition can be written in terms of the heat …