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Questions tagged [elliptic-equations]

For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).

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13 views

Convergence of Sequence of Solutions to Elliptic Equation

Consider the standard uniformly elliptic equation on a domain $\Omega \subset \mathbb{R}^d$: $$ \mathrm{div}(A(x)\nabla u) = f $$ for $u \in H^{1}(\Omega)$ , $f\in H^{1} (\Omega)$, $a_{ij}(x)$ ...
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2answers
32 views

A (nontrivial) nonlinear Poisson equation

Let $U \subset \mathbb R^3$ be bounded, with smooth boundary. Does the equation $$ -\Delta f = e^{-f} \text{ in } U, \quad f = 0 \text{ on } \partial U$$ has a solution? Is there a name for this ...
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0answers
23 views

$L^\infty$ boundedness for solution of elliptic PDE with Neumann BC

On a bounded smooth domain in $\mathbb{R}^n$ consider the equation $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$where $k>0$ is a constant and $f \in L^2(\Omega)$. We know that $u \in H^2(\Omega)$. ...
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0answers
20 views

Are Kolmogorov partial differential equations degenerate elliptic?

Let $\mu : \mathbb{R}^d \rightarrow \mathbb{R}^d $ and $\sigma: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d \times d} $ be smooth and Lipschitz continuous. Furthermore, let $\varphi : \mathbb{R}^d \...
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0answers
10 views

$L^2$-norm of the solution of a 2-order elliptic nonlinear equation

When $1<p<2^*$, $E>0$ is a constant, assuming the positive solution of equation below is $U_E$ $$ -\Delta u + Eu - |u|^{p-1}u =0, ~~~~x\in \mathbb R^n (n\ge1) $$ as I read in Evans' book, the ...
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0answers
29 views

Minimal surface with a “flat curve”

Let $\Sigma^2 \subseteq \mathbb{R}^3$ be a complete minimal surface and let's assume that there exists a smooth regular curve $\gamma \colon I \to \Sigma$ such that $K(\gamma(t)) = 0$ for all $t \in ...
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0answers
23 views

Global elliptic regularity theory on $\mathbb{R}^n$ or interior estimates for elliptic pde when $p \neq 2$

Let $p \in (1.\infty)$ and $a_{ij},b_j,c \in C^\infty(\mathbb{R}^N)$ be the bounded coefficients of the elliptic second order differential operator $$[Au](x) = - \mathrm{div}(A(x) \nabla u) + \left\...
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1answer
41 views

Uniqueness of solution for Dirichlet and Neumann problem [closed]

How to prove the uniqueness of the Dirichlet and Neumann problems for the equation $$ \Delta u - ku=0$$ in a bounded domain $D \subseteq \mathbb{R}^2$, where $K$ is a positive constant? What happens ...
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0answers
12 views

How does the conservation law for the Schrodinger's equation change with respect to its coefficient?

In Tao's book, page 84, he stated the conservation law for the linear Schrodinger's equation $i \partial_t u+ \frac{1}{2}\Delta u = 0$. But there are other resources treating the equation as $\...
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20 views

Laplace equation on rectangles

I’m reading Pinchover and Rubinstein. On page 189 example 7.21 it tries to solve a Laplace equation in the rectangle $0<x<b , 0<y<d$ subject to the Dirichlet boundary condition:$$u(0,y)=f(...
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10 views

Generalization of Pohozaev identity

Is it possible to generalize Pohozaev identity (and the related non-existence result) to functionals of the form $$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \frac{1}{2^*}|u|^{2^*}...
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0answers
23 views

Eigenvalue of nonlinear elliptic equation.

For any $u \in H^2(\mathbb R^n)$, consider operator $$ Lu = -\frac{1}{2}\Delta u + u - 3u_0^2u $$ where $u_0$ is a solution of $$ -\frac{1}{2}\Delta u + u - u ^3 =0 $$ how to show $L$ has only one ...
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0answers
28 views

How to show minimal surface equation is a non-uniform elliptic equation

The minimal surface equation is $$ \sum D_i \left(\frac{D_i u }{\sqrt{1+ |Du|^2}}\right)=0 $$ How to show it is a non-uniform elliptic equation ? What I try: $$ D_i \left(\frac{D_i u }{\sqrt{1+ |Du|^...
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1answer
35 views

$-\Delta u=f$ in $L^1$ but $u_{x_ix_j}$ not in $L^1$ ($i\neq j$)

I want to show the following function is a counterexample for Poisson equation with $L^1$ RHS but its solution is not in $W^{2,1}$: Let $n\geq3$ and $x\in\mathbb{R}^n$. Let $f(x)=|x|^{-n}(\log|x|)^{-...
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1answer
128 views

Laplacian with Integral BC(s)

I want to solve the three-dimensional laplacian $$\nabla^{2} T = 0$$ where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ defined on $...
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0answers
26 views

Laplace equaion with integral source terms

I have the following coupled PDEs: \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - ...
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0answers
38 views

On the solvability of the Dirichlet Problem $\Delta u =f$ for $f$ locally Holder continuous and $L^p$ for $p>n/2$

It's well known that if $\Omega$ is a bounded set and $f$ is locally Holder continuous on $\Omega$ and bounded, then $u = \int_{\Omega} \Gamma(x-y)f(y) \ dy$ is a classical solution to $\Delta u=f$, ...
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0answers
17 views

Does the linearized equation of uniformly elliptic equation maintain uniformly elliptic?

I want to consider whether the linearized equation of (nonlinear) uniformly elliptic equation maintains uniformly elliptic or not. By definition, for the original uniformly elliptic equation:$$ F(D^...
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1answer
70 views

Show that solution has range $[-1,1]$ [duplicate]

Let $u$ be a solution of $\Delta u=u^3-u$ on a bounded domain $\Omega$. I want to show that $u\in[-1,1]$ throughout $\Omega$ and find out if $u$ can attain the values $-1,1$. This is an exercise from ...
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1answer
26 views

Boundary problem with elliptic 2nd order pde

I am reading some notes about harmonic functions and associated boundary problems and I came through a proposition stating that the problem $u_{xx}+u_{yy}=0$ $u(0,y)=u(1,y)=u(x,1)=u(x,0)=0$ , where $...
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0answers
50 views

Harnack Inequality for nonnegative subsolutions to uniformly elliptic PDE

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$,where $A = A(x)$ is uniformly elliptic. The proof outline I am following is ...
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3answers
148 views

How to solve a second order partial differential equation involving a delta Dirac function?

In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function: $$ a \, \frac{\partial^2 w}{\partial x^2} + b \, \frac{\partial^2 w}{\...
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0answers
15 views

Regularity for weak solution of Poisson problem in a rectangle

Let $\Omega=(0,1)^2$. Let $u$ be a weak solution of $\Delta u=f$ con $f \in L^2(\Omega)$ e $u \in H^1_0(\Omega)$. I would like to prove that $u \in H^2(\Omega)$. I know that $u \in H^2_{loc}(\Omega)$ ...
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0answers
34 views

Question about rational points on elliptics curves.

I have the following question about this lemma, Where $A,B\in\mathbb{F}$. I tried to prove that $\mbox{Norm}_{R/F}(\alpha-\beta\theta)=f^{hom}(\alpha,\beta))$, but I don't see, why this is correct. ...
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1answer
208 views

Three Coupled PDE. Is my approach in the right way?

I have the following three PDEs \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - \...
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1answer
55 views

Non-singular elliptic curve parametrization

It is known that some singular elliptic curves can be expressed with parametric equations. For example : $y^2=x^3$ can be parametrized with $x=t^2$ and $y=t^3$ $y^2=x^3+x^2$ can be parametrized with $...
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1answer
33 views

Elliptic Regularity on Convex Domain

In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds) However, it seems to me that the polygonal ...
4
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0answers
99 views

Partial differential equation with a nowhere differentiable boundary

Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. ...
1
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1answer
54 views

Finding a solution of a PDE, which is growing in time

I've been dealing with a problem about partial differential equations for a while and hope you can help me. Let be $\omega, c>0$ and $h: [0,a]\rightarrow \mathbb{R}$ regular ($a>0 \in \mathbb{R}...
2
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1answer
38 views

About positivity of a solution to a sub-critical semilinear elliptic problem

Let $\Omega$ be a non-empty open bounded subset of $\mathbb{R}^N$ with $N\ge 3$. Let $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function such that $\exists a,b>0, \exists\...
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1answer
22 views

Decay of Dirichlet exterior problem

Assuming $\Omega \subset \mathbb R^n, (n\ge 3)$ is a domain. Consider the Dirichlet exterior problem \begin{align} &\Delta u = 0 ~~~~x\in \mathbb R^n/\Omega\\ &u|_{\partial \Omega} =1 \tag{1}...
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2answers
41 views

Laplace equation with the Robin's boundary problem

$\textbf{Problem}$ Let $\Omega$ be an open, bounded and connected subset of $\mathbb{R}^n$. Suppose that $\partial \Omega$ is $C^{\infty}$. Consider an eigenvalue problem \begin{align*} \begin{...
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1answer
95 views

Estimate of a weak solution in a nonhomogeneous equation

$\textbf{Problem}$ Let $\Omega \subset \mathbb{R}^n$ be open, bounded and connected with $\partial \Omega\in C^1$. For each $i,j=1,\cdots,n$, assume that $a_{ij},b_i,c \in L^{\infty}(\Omega)$ (real ...
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1answer
62 views

Proof of Lemma 2 in Jürgen Moser's Proof of the Nash / De Giorgi / Moser Theorem

I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations" (which can be found here: https://onlinelibrary.wiley....
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1answer
29 views

Neumann problem with zero average sobolev space

Let us considère the following Laplace-Neumann problem $-\Delta u=0$ with homogenuous boundary condition of type neumann, i:e $\frac{{\partial u}}{{\partial n}} = 0$. The variational formulation is ...
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0answers
25 views

Verify $u=sech|x|$ solute $-\frac{1}{2}\Delta u + \frac{1}{2}u - u^3 =0$

I try to verify $u=sech|x|$ is a solution of $-\frac{1}{2}\Delta u + \frac{1}{2}u - u^3 =0$. But there is a little error. What I try: \begin{align} &\frac{\partial |x|}{\partial x_i}=\frac{x_i}...
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0answers
18 views

Stretched elastic band shape

I have written a python code to obtain the minimum energy of a network of beads connected to one another via springs (all with elastic constants $k=1$). The rest length of each spring is zero, to ...
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0answers
15 views

Reference about eigenvalue of Kirchhoff type operator

Assume $u_0$ is a positive radial symmetric nontrival solution of \begin{align} -A(||\nabla u||^2)\Delta u + \lambda u - |u|^2u=0 \tag{1} \\ u\in H^2( R^n)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{...
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0answers
24 views

Finite Element formulation of mixed BVP of Variational Problem

Suppose we are given the followin where $f$,$u$, $g$ are given functions: $-\Delta u = f$ in $\Omega$ $u=u_o$ on $\Gamma_1$ $\frac{du}{dn}=g$ on $\Gamma_2$ So in order for me to form the ...
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1answer
48 views

A coupled problem of elliptic equations.

Let $Ω$ be an open bounded on class $C^1$. We consider, on $Ω$, the problem with the following limits: $(P)\quad \left\{ \begin{array}{lcc} -\Delta u_1 +\alpha u_2=f_1 & \text{on ...
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0answers
62 views

Semilinear elliptic equation $\Delta u = P(u)$ with $P$ being polynomial of degree 3

Suppose that $B_1 \subset \mathbb{R}^3$ and $P(u)$ is a polynomial with degree 3. If $u \in W^{1,2}(B_1)$ is a weak solution of $$\Delta u = P(u) \text{ in } B_1,$$ then can we obtain the ...
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2answers
81 views

Eigenvalue of a given operator

If $u_0$ is a positive radial symmetric nontrival solution of $$ -\frac{1}{2}\frac{d^2u}{dx^2}+\lambda u -u^3=0 $$ Then how to show $-3\lambda$ is a eigenvalue of $$ Lu=-\frac{1}{2}\frac{d^2u}{dx^2}+...
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0answers
26 views

Question on optimal regularity for the elliptic Neumann problem

I 'm reading a paper at the moment and I have a really hard time understanding the following: Let $U \subset \mathbb R^3$ be open, bounded and connected with a $C^2-$ regular boundary $\partial U$...
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1answer
54 views

Higher regularity for solutions of elliptic equations

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$. Let $f\in L^\infty(\Omega)$. For the problem $$-\Delta u=f\mbox{ in }\Omega\\ ~~~~~~~~~u=0\mbox{ on }\partial\Omega,$$ one could seek solutions in ...
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1answer
32 views

Existence of Positive Solutions to Constrained Linear Elliptic Second-Order PDE

Consider the elliptic second-order PDE on some bounded domain $D$ (in any dimension) $-\Delta_D u + \alpha u = 0$ subject to the constraint $\gamma^i \nabla_i u + \beta u = 0$, where $\Delta_D$ is ...
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2answers
100 views

find the locus of the vertices of equilateral triangle circumscribing the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ [closed]

My try : I am confused in this question , I have only tried questions who says to find the locus of a point but here we have to find locus of three points and there are only two relation , seems ...
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0answers
22 views

Neumann Laplace eigenfunctions

Let $u_k, u_m$ be two Neumann Laplace eigenfunctions on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, corresponding to eigenvalues $\mu_k, \mu_m$ respectively. ...
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0answers
23 views

Eigenfunction of Dirichlet Laplacian on smooth domain in $\mathbb{R}^n$

I was reading about eigenfunctions of the Dirichlet Laplacian on bounded domains $\Omega \subset \mathbb{R}^n$. It seems that such eigenfunctions are real analytic in the interior of $\Omega$ and ...
0
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1answer
25 views

Elliptic PDE- solution in weak sense

I looking for some literature, where I can find strategy how to solve elliptic PDE in weak sense - define notion of weak solution. Can anyone recommend me something ? Or some links to solved examples. ...
0
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1answer
142 views

what is the equation of a rotated ellipsoid?

I used this equation for rotated 2D ellipse: $$\frac{(x\cos\theta+y\sin\theta)^2}{a^2} +\frac{(x\sin\theta-y\cos\theta)^2}{b^2} =1$$ $a$, $b$ are major-axis and minor-axes length of the ellipse and $...