# 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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### Asymptotically periodic potentials

Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials?

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53 views

### On boundary-value problems for differential forms on a manifold

Let $M$ be a simply-connected $d$-dimensional Riemannian manifold with boundary (for simplicity assume a ball). Consider the boundary value problem for $\omega\in\Omega^k(M)$,
$$
d\omega = \alpha
\...

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35 views

### Reference request: application of Minty - Browder theorem to a quasilinear elliptic pde problem

I am working on a research and I'm having a hard time understanding how to use Minty-Browder theorem on a quasilinear elliptic pde problem with a Neumann boundary condition. I am really looking for a ...

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30 views

### The order of regularity improving for elliptic operator with rough coefficients

Let $\frac12<\alpha<1$ and let $L(x)$ be a first order overdetermined elliptic operator with coefficients in $C^\alpha$. We means $L(x)=\sum_{j=1}^nA_j(x)\partial_j$ where $A_j:\mathbb R^n\to\...

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171 views

### Why does this PDE have a solution?

Let $(M^3,g)$ be a compact $3$-manifold with boundary and let $\Sigma$ be a surface such that $\partial M \cap \Sigma = \partial \Sigma$ and the intersection is orthogonal ($\Sigma$ is a free-boundary ...

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69 views

### Proving Harnack for viscosity solutions

In reference to Theorem 4.8 'Fully Nonlinear Elliptic Equations' by Caffarelli and Cabre. We are asked to prove two things given $u\in \overline{S}(\lambda,\Lambda,f), u\geq 0$ in $Q_1$, and $f$ is ...

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103 views

### Minimal assumptions such that the solution of Poisson equation is $C^2$

Take a weak solution $u$ of the Poisson equation on $\mathbb{R}^d$
$$ \Delta u = f $$
By standard elliptic regularity theory we have (for some $\alpha\in (0,1]$) $f\in C^{0, \alpha}_{\text{loc}}(\...

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107 views

### Optimal Sobolev regularity for $(-\Delta)^{-s}$ on domains

In the paper Chen, Huyuan; Véron, Laurent, Semilinear fractional elliptic equations involving measures, (J. Differ. Equations 257, No. 5, 1457-1486 (2014). ZBL1290.35305) in the proof of Proposition ...

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67 views

### Partial regularity of harmonic maps into spheres

Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...

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190 views

### Elliptic Regularity with Gibbs Measure Satisfying Bakry-Emery Condition

Consider $\mathbb{R}^d$ with Gibbs measure $d\mu=Z^{-1}\exp(-V(x))dx$, where the potential $V(x)$ is strongly convex ($\nabla^2 V(x) \ge \lambda Id $). We can assume the regularity of $V$ is as good ...

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53 views

### Small energy implies a lifting $\rho e^{i\theta}$

Set $T^N$ the $N$-dimensional torus and $u\in H^1(T^N,\mathbb{C})$. Can I say that if the energy$$\int_{T^N}|\nabla u|^2 +\frac12\int_{T^N}[1-|u|^2]^2$$ is small enough (let say lower than some $\...

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38 views

### Zero in the spectrum of an elliptic second order operator

This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator
but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...

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51 views

### Eigenvalue problem with non homogeneous Robin boundary

I am interested in the eigenvalue problem
$$
\left\lbrace \begin{array}{ll}-\Delta u=\lambda u\textrm{ in }\Omega,\\\frac{\partial u}{\partial \nu}+\alpha u=g\textrm{ on }\partial\Omega. \end{array}\...

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69 views

### Spectrum of a linear elliptic operator

In the paper in quantum fields theory by
Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19;
in Section 3 the author makes the following claim from PDE and ...

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147 views

### Construction of elliptic equation with Neumann boundary condition from a minimization problem

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.
Let $B=B_1 \subset \mathbb{R}^3$ and $E : H^1(B) \to \mathbb{R}$
$$E(...