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Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text …
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8 votes
1 answer
446 views

On critical points of harmonic functions

Let $u \in C^{\infty}(\mathbb R^3)$ be harmonic. Suppose that $u$ has no critical points outside the unit ball but that it has at least one critical point inside the unit ball. Does it follow that $u$ …
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  • 4,115
8 votes
1 answer
520 views

Dirichlet-to-Neumann map on Lipschitz domains

Let $\Omega$ be a bounded domain with a Lipschitz boundary. Consider the Dirichlet-to-Neumann map $\Lambda:H^{\frac{1}{2}}(\partial \Omega)\to H^{-\frac{1}{2}}(\partial \Omega)$ defined via $$ \langle …
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  • 4,115
5 votes
1 answer
190 views

Mean value principle reversed

Suppose that $\Omega \subset \mathbb R^3$ is a domain with smooth boundary $\partial \Omega$ and suppose that $0\in \Omega$. Given any $f \in C^{\infty}(\partial \Omega)$ let $u^f$ denote the unique h …
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  • 4,115
4 votes
0 answers
81 views

On the convergence of the spectral decomposition of a harmonic function

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$, …
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  • 4,115
4 votes
2 answers
356 views

Nontrivial invariant transformations for heat equations

It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by $$ v(r,\theta) = u(\frac{1}{r},\theta)$$ is also harmonic for $r>0$. Note that the Kelvin tran …
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  • 4,115
3 votes
0 answers
65 views

Elliptic equations in semi-infinite strips

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good …
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  • 4,115
3 votes
1 answer
153 views

Foliation by Asymptotic lines

Suppose $(M,g)$ is a compact Riemannian manifold with boundary. I am interested about existence of surfaces $\Gamma$ embedded in $M$ with the following property: $\Gamma$ is foliated by geodesics (he …
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  • 4,115
3 votes
1 answer
1k views

Continuation (extension) of harmonic functions

Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the …
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  • 4,115
3 votes
1 answer
113 views

Boundedness of solutions to a semilinear PDE

Let $D$ be the unit disk in $\mathbb R^2$ centered at the origin. Given any $\lambda \in \mathbb R$, let $u_\lambda$ be the unique solution to the semilinear elliptic equation $$ -\Delta u + u^3=0 \qu …
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  • 4,115
3 votes
1 answer
161 views

On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$ and let $G: L^2(\Omega)\to H^2(\Omega)$ be the …
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  • 4,115
2 votes
0 answers
76 views

Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator

I would appreciate any answers or even references for the following problem. Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the metri …
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  • 4,115
2 votes
0 answers
153 views

unique continuation in a disk

Let $D$ be the unit disc in $\mathbb R^2$ centered at the origin. Let $w \in C^{\infty}_c(D)$ satisfy $$ (1-r^2)^2\Delta w +w =0.$$ Prove that $w \equiv 0$.
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  • 4,115
2 votes
0 answers
57 views

Uniqueness for a certain semilinear equation

Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation $$ \begin{aligned} \begin{cas …
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  • 4,115
2 votes
2 answers
131 views

Density of traces of solutions to an elliptic equation

Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus D_ …
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