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        Questions tagged [elliptic-pde]

        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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        Green’s function on upper unit half ball [on hold]

        Suppose we have this region in $\mathbb{R}^3$ $$ (x,y,z) \in \mathbb{R}^3, x^2+y^2+z^2<1, z>0$$ How to find Green’s function for this domain? And how it is possible to write the integral ...
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        Neumann functions of Poisson problem [on hold]

        On page 219 of “Pinchover & Rubinstein” it is trying to find a function which is called Neumann function for $$ \Delta u= f, D$$ $$ \partial_n u= g, \partial D$$ It introduces $$h(x,y;\zeta,\eta)$...
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        126 views

        Eigenfunctions of elliptic equations

        Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
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        A particular semi-linear equation on Riemannian manifolds

        Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
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        1answer
        56 views

        Stability of minimal hypersurface with flat directions

        Let $\Sigma^n \subseteq \mathbb{R}^{n+1}$ be a minimal complete hypersurface. Let's consider $\bar{\Sigma} := \Sigma \times \mathbb{R}^k \subseteq \mathbb{R}^{n+k +1}$. Clearly $\bar{\Sigma}$ is a ...
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        PDE on an open ball with prescribed value on some open subsets

        Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
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        “Brownian motion” related to the $p$-Laplace operator

        The link between the Brownian motion and the Laplace operator is well-known. What stochastic process plays an analogous role with respect to the $p$-Laplace operator?
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        Pohozaev identity and related non-existence result for a nonlinear problem

        Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional $$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...
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        31 views

        Different solution sets of Partial Differential equations

        Consider Laplace equation $?^2u=0$. We can find a set of solutions for that by assuming $u=f(x)g(y)$. Also we can find another set of solutions by assuming $u=f(x)+g(y)$ that is not the same as the ...
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        Ellipticity-type condition

        An elliptic operator $L=\mathrm{div}(A(x)\nabla u)$, is called uniformly elliptic if $$C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$$ If $A$ depends also on $u$, what is the condition $$C^{-1} + C^...
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        1answer
        104 views

        Oscillation and Holder continuity

        Where can I find a proof of the follwing fact? If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \...
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        0answers
        59 views

        Laplacian variational problem with asymptotically quadratic term

        Consider the functional $$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$ where $\Omega$ is a bounded smooth domain. The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...
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        77 views

        Singular integral operators and PDEs

        What is the relation between the notion of singular integral operators and partial differential equations? I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
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        1answer
        173 views

        Local “boundary comparison principle” for harmonic functions

        Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
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        0answers
        91 views

        Green function of a fractional operator

        What is the Green function of the following operator with homogeneous Dirichlet boundary condition? $$(-\Delta)^s - k \frac{u}{|x|^{2s}} \quad (k\ge 0) $$

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