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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.

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        293 views

        Does current follow the path(s) of least (total) resistance?

        Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...
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        vote
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        72 views

        Solutions to $\Delta u\ge u^2$

        Let $(M,g)$ be a complete Riemannian manifold. Suppose that $u$ is a nonnegative solution to $\Delta_gu\ge u^2$. Does it follow that $u$ must be identically 0? I know that the answer to above ...
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        35 views

        Interpolation inequalities involving mean curvature operator

        Are there any interpolation inequalities (for example, of Gagliardo-Nirenberg type) involving the mean curvature operator $$\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)$$ (in any ...
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        votes
        0answers
        50 views

        Reference on elliptic obstacle problem that covers the material in the lecture notes by Caffarelli

        Can you recommend a modern, self-contained, readable reference that covers (approximately) the results on elliptic obstacle problem that are covered in the lecture notes by L. Caffarelli (Scuola ...
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        votes
        1answer
        73 views

        Generation of strictly contraction Semigroups

        Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...
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        Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

        Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\...
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        votes
        2answers
        216 views

        Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

        Where can I find a (readable and self-contained) proof of the following result? Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
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        votes
        0answers
        138 views

        DeGiorgi oscillation lemma

        Where can I find a proof of the following result? Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$ where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(...
        3
        votes
        1answer
        115 views

        Global regularity for Neumann problem

        Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
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        63 views

        Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary

        In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669 Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...
        2
        votes
        0answers
        38 views

        Bessel decay for nonhomogeneous PDE

        I'm interested in the following nonhomogeneous PDE $$ (\Delta-k^{2})u=-g $$ on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
        4
        votes
        0answers
        257 views

        Spectral Gap of Elliptic Operator

        Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled? The boundary condition is that the ...
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        votes
        3answers
        197 views

        about the Hausdorff dimension of Removable singularities of PDE

        There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ...
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        votes
        0answers
        68 views

        Green functions for circular sectors

        I would like to solve the Dirichlet boundary value problem $$ (\Delta-k^2)u=0 \ \ \ \text{in $\Omega$} \\ u=f \ \ \ \text{on $\partial \Omega$} $$ where $\Omega$ is an infinite circular sector of ...
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        votes
        2answers
        219 views

        Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

        There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs, where $P_k$ is given by \begin{equation} \begin{aligned} P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...

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