# 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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### Applications and motivations of resolvent for elliptic operator

Let $A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix satisfying ellipticity condition, that is \begin{align} \mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2 \...
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### An inequality involving supremum over the boundary

Apologies if this is not fitting for the site, I will remove the post if requested. I am studying elliptic PDE from the book of Chen and Wu, Second Order Elliptic Equations and Systems. I am having ...
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### The spectrum of Laplacian operator

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$. For $f\in L^2$, it is well known that we have a unique solution $u\in H_0^1(\Omega)$ by using Lax-Milgram theorem for the Dirichlet problem ...
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### Poisson Kernel and solution formula for fractional elliptic problem

$$k (-\Delta)^s u + u = 0, \qquad x \in U, \\ u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,$$ with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
1 vote
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### Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate. I am reading a paper of Brezis and Oswald about existence ...
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### Kernel of the Laplacian + a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
1 vote
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### Problems arising from a paper on the radial symmetry of the global solution of semilinear PDE $\Delta u+f(u)=0$ in $\Bbb{R}^{n}$

I am reading the paper  by Congming Li. I want to talk about the typical case that the author gives as follows (, §1, pp. 590-): In this section, we study positive solutions of the following ...
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### A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture

[This question is looking at the paper Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/...
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### Differentiability of a weak solution

Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...
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### Would you help me to find this expression?

I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate $$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$ Where $\Omega$ is an ...
1 vote
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### Target space of Green's operator on $L^p$-differential forms on closed manifolds

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott , i.e., with a finite atlas $\mathcal{A}$ so that for ...
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### Existence of ground state solutions for the critical exponent

I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$. The authors show that the above equation has a unique positive ...
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### Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$

Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
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### The behavior of $\nabla u$ on the boundary for Poisson equations

Let $\Omega$ be a bounded domain with smooth boundary. Consider the Poisson equation \begin{eqnarray} -\Delta u&=&f\text{ in }\Omega\\ u&=&0\text{ on }\partial\Omega \end{eqnarray} ...
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### Non-existence of rapidly decaying solutions of certain elliptic semilinear equations

Consider the equation $$-\Delta f+mf+\lambda f^p=0$$ on $\mathbb{R}^d$, where $d>2$,$m>0$, $p>1$ is integer, and $\lambda \in \mathbb{R}$. Are there any known results regarding the non-...
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### $C^{1,\alpha}$ estimate for Newton potential of $L^\infty$ function

Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then $$u(x) = \int_\Omega \log |x-y| f(y)\ dy$$ ...
1 vote
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How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation? Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$-\Delta u + v \cdot \nabla u = 0 ... 1 vote 0 answers 52 views ### Boundary estimates for elliptic systems Let  \Omega\subset\mathbb{R}^d  is a  C^{1,\eta}  domian with  0<\eta<1 . Assume that  A(y)=(a_{ij}^{\alpha\beta}(y))  is a matrix valued function, where  1\leq i,j\leq d  and  1\leq\... 1 vote 0 answers 41 views ### Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter Let \sigma \in[0,1],we consider following series of linear partial differential equations related to the parameter \sigma,for example$$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \...
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Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial ...