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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 …

$\newcommand{\R}{\mathbb R}$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 $ …

No, we cannot. Formally, $\varphi$ is the eigenfunction of the unbounded operator $L_s$ on $L^2(\Omega)$, defined initially by $$ L_s u(x) = (-\Delta)^s u(x) \qquad \text{for } x \in \Omega , $$ where …

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 …

I do not there is a strong relation between the two notions in the general case. Curvature is obviously a local object. On the other hand, the behaviour of the first Dirichlet eigenfunction near a bou …

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^ …

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 …

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 …

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 …

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) …

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 …

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 $ …

(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 …

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 …

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 …