# Tagged Questions

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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### Continuity + $H^1$ + Laplacian control $\implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...
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### Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...
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Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$. For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...
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### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}.$

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
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### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...
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### Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says: Suppose $u \in C^2(\Omega)\cap C(\overline{\Omega})$ satisfies $-\Delta u \geq 0$ in $\Omega$. ...
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### Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
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### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...
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### Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
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### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} ‎\Delta^2u=‎‎\lambda ‎‎\dfrac{u}{|x|^4}‎‎+u^{‎p}‎ & \mathrm{in}‎\hspace{...
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### Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...
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### Cea's lemma and norms

I would like your help understanding this article. Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing. Things I do not ...
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Let us see the heat kernel $$k(x,t)= \begin{cases} (4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\ 0,t\leq0. \end{cases}$$ It is easy to see that $k\in C^{... 0answers 187 views ### One parameter family of elliptic equations Consider the following 2nd order nonlinear elliptic equation on$\mathbb{R}^n\$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = N(\... 0answers 83 views ### Examples for differential operators first order Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain \Omega\subset\mathbb{R}^n with certain regular bondary and a function ... 0answers 73 views ### biharmonic equation with L^1 data and Navier Condition I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ... 2answers 271 views ### What can we say about the boundary of the level set of a Sobolev function? I'm a beginner of the area of free boundary problem. Let me first give some background: \Omega \subset \mathbb{R}^n is an open connected set, and locally \partial \Omega is a Lipschitz graph. ... 2answers 150 views ### References for non-zero boundary value problem I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity. I studied always, following Evans book "PDE", the case with ... 0answers 106 views ### Regularity of weak solution I have also posted the question here. Let me explain what difficulties I have. In fact, one may write $$\partial_1(f-\partial_1 u)=0$$ in \Omega. Then one may have the ... 3answers 194 views ### Limits for eigenvalues for the Dirichlet Laplacian If \Omega is a bounded domain in \mathbb{R}^n, let \lambda(\Omega) be an eigenvalue of the problem$$ \begin{cases} -\Delta u=\lambda u & \mbox{in }\Omega\\ u=0 & \mbox{on }\partial\...
I’m interested in solving nonlinear elliptic boundary value problems of the type $$-a\Delta u + f(u) = 0,$$ $$u|_\Gamma = u_0$$ by Newton’s method when its convergence is global and monotonic. ...