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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8 views

Reduction to Fixed point theorem in PDE

Let $Q$ be an operator of the form: $$Qu:=a^{ij}D_{ij}u +b$$ where $a^{ij}$ and $b$ are functons of $(x,u,Du)$ and $u$ is a function of $x$. To reduce the existence of the solution $u$ to the fixed ...
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13 views

Moser Iteration for Laplacian with Hardy potential

I am reading the following proof of Cao-Yan's 2010 CVPDE Paper. It's a property for solutions of Laplace equation with Hardy potential $|x|^{-2}.$ The space dimension $N \geq 3.$ Their proof is in the ...
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1answer
70 views

Maximum Principle for Minimal Surface Equation with Dirichlet Boundary Condition

I'm an undergraduate student and I'm currently reading a classical paper for my final project for the course differential geometry on the Bernstein problem of minimal surfaces, namely, the paper: ...
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29 views

solving nonlinear second order differential equation

I was stuck in solving the following nonlinear second order differential equation $$ y''=\dfrac{1}{y}. $$ I try to use the Laplace transform to solve it but not work especially when dealing with the ...
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1answer
42 views

Physical Interpretation of degenerate Laplacian

Consider the equation $$-\frac{\partial^2u}{\partial x^2}=-\text{div}\left(\begin{pmatrix}1&0\\0&0\end{pmatrix}\nabla\right)=0\mbox{ in }\mathbb{R}^2.$$ Solving the equation, we can conclude ...
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1answer
68 views

A PDE Exercise using the Maximum Principle

I have an exercise from the PDE class: Suppose u is a smooth solution to $$ \left\{ \begin{aligned} Lu:= \Delta u + cu = \vert \nabla u \vert^2 \quad \text{in}\ \Omega, \\ u = 0 \quad \text{on}\ \...
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14 views

Stochastic Connection Between Elliptic and Parabolic PDE

This question is something on the lines for what I am looking for, Estimates for parabolic vs elliptic PDE. For certain elliptic PDE's, we can show the stochastic representation of a solution to the ...
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42 views

Existence of a solution for the discrete pde $\nabla a \nabla f = g$ in $\mathbb{Z}^d$ with $g$ periodic.

I would like to know how does one solve the problem $\nabla^* a \nabla f =g$ in $\mathbb{Z^d}$ with $f(0)=c$ where $$ \nabla^* a \nabla f(x) = \sum _{y \sim x} a(x,y)(f(y)-f(x)) $$ with $a(x,y)=a(y,x)...
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1answer
46 views

Solution of Elliptic PDE

I'm trying to solve the following problem. Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded set. Consider a differential operator defined, for $u \in C_0^{\infty}$ as follows: $$ Lu(x):=\sum_{i=...
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14 views

Differentiability of parameter in parameter-dependent equation

Let $A(t)$ be a positive-definite matrix that depends on a parameter $t$. Fix $f \in H^{-1}(\Omega)$. Suppose that the function $u=u(t) \in H^1_0(\Omega)$ is defined as the unique solution of $$\int_\...
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1answer
40 views

Well-posed Elliptic boundary value problems

I've been taught that elliptic boundary value problems (BVPs) are well-posed i.e. a solution exists, it is unique, and it depends continuously on the boundary data. Why then does the following 2d ...
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35 views

Second order linear elliptic PDE

Is anyone aware of how to find an analytic, exact, general solution to the following elliptical PDE? $$ 2\partial_{x}\partial_{y}f=x\partial_{x}f+y\partial_{y}f $$ The techniques I've seen for ...
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16 views

Semilinear maximum principle

I am considering the problem $$ -\nabla\cdot(a(x)\nabla u) + f(u) = C, $$ posed on a rectangular domain with periodic boundary conditions. $a(x)$ is a bounded non-negative function, $C$ is a positive ...
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1answer
28 views

Why distance function differentiable for sufficiently smooth boundary?

Suppose that $\Omega$ is bounded domain in $\mathbb{R}^n$ with $C^k$ boundary. Why is it that for points sufficiently close to the boundary the distance function $d = d( \cdot, \partial \Omega)$ is $C^...
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25 views

What is k(T) related to the elliptic curve and how this can be evaluated?

In this paper, The 1729 K3 Surface, the link is here https://arxiv.org/abs/1510.00735, Ken Ono writes In particular, parametrized solutions to $X^3 + Y^3 = Z^3 + W^3$ give us families of elliptic ...
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55 views

Maximum principle for elliptic equation in exterior domain

I have a question on maximum principle for elliptic equation in exterior domain. Suppose that $u$ is infinitely differentiable in $\mathbb{R}^n$ and bounded in $\mathbb{R}^n$. I want to prove that ...
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27 views

The Laplace equation and an auxiliary boundary problem

I have the following problem that the teacher proposed to us in a task. The statement is more or less as follows: The object of this problem is the study of a strategy of resolution of type ...
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1answer
16 views

Parabolic to Elliptic

Let $u:\Omega\times(0,\infty)\to(0,\infty)$ satisfies the heat equation $$ u_t=\Delta u, $$ then does the function $v(x)=u(\cdot,t)$ defined for every fixed $t$ satisfy the equation $$ \Delta u=0 \...
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28 views

Multiplication by $n$ formula on elliptic curves

Let be $E:y^2=x^3+ax^2+bx+c$ an elliptic curve, with $a,b,c\in\mathbb{Q}$, and $P=(x,y)\in E$. I would like to show: $\forall n\in\mathbb{Z}\hspace{0.2cm}\exists \phi_n,\psi_n\in\mathbb{Q}[X]: x(nP)=\...
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21 views

Weak maximum principle for eliptic PDEs proof question

I have been given a proof for the weak maximum principle and I do not understand the logic in one of the steps. Let $a_{ij}$ be syymetric then: $$Lv=\sum_{i,j=1}^N a_{ij}(x)\partial_{x_i}\partial_{...
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32 views

$\Delta u = u$ implies $u$ infinitely differentiable, without Sobolev techniques

Suppose $u:\mathbb{R}^n \to \mathbb{R}$ is a $C^2$ function which satisfies $\Delta u = u$. Can we prove that $u$ is $C^{\infty}$ without using the elliptic regularity theorem or any machinery of ...
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24 views

Bootstrapping For Elliptic Operators in Holder Space

Do we have a proposition in the following form: L is an elliptic operator on $\Omega$. If we have $Lu = v$ with $v\in C^{k-2,\beta}(\Omega)$ and $u\in C^{k}(\Omega)$. We actually know that $u\in C^{k,...
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110 views

Find a weak solution of $- \Delta u + u = h(x) |u|^{q - 2}u$ in $\Bbb{R}^N$, $u \in H^1 (\Bbb{R}^N)$ - Is my reasoning correct?

The following is a problem in a text on Critical Point Theory (in Portuguese) I am reading. I would like to know if my reasoning is correct. One observation: the Ekeland Variational Principle cited ...
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Why holder space is more appropriate than $C^2(\Omega)$ for elliptic equation

The Holder space has been extensively used in elliptic equation. I am just wondering why it is a more proper choice than $C^2(\Omega)$ in elliptic theory
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5 views

PDE and continuation method / Newton's method

I want to solve the nonlinear elliptic equation $$ \begin{cases} \Delta u+ f(u)=g \;\; \text{in} \; \; (0,1)\\ u(0)=u(1)=0 \end{cases} $$ where $f\in C^1(\mathbb{R})$ and $g\in L^2(0,1)$ My idea, ...
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1answer
11 views

Why is this function a local minimum of this integral functional? (related to elliptic PDEs)

This is actually a step in a bigger proof about regualarity properties of elliptic systems, but I'll write only what I don't understand. Let $L: \mathbb{R}^{m \times n} \to \mathbb{R}$ be $C^2$ and ...
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29 views

Elliptic curve in finite field, which solution to choose?

This is the elliptic curve I'm working with in the finite field of mod 37: $$ y^2 = x^3 - 5x + 8 $$ While trying to generate the set of points in it I noticed that, for $x = 5$, $$ y^2 = 5^3 - 5\...
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15 views

Norm in $H_0^1(\Omega)$ for generalized Nehari

In this paper by Szulkin and Weth, when using the generalized Nehari manifold in section 3, they claim that one can introduce an inner product such that the induced norm satisfies $$\lVert u^+\rVert^2 ...
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1answer
31 views

A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ ...
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1answer
59 views

Invariant interval for ODE implies invariant interval for parabolic PDE

Let $u$ solve $u_t-\Delta u = f(u)$ on $[0,T] \times \Omega$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $u=0$ on $\partial{\Omega}$. Also, let $\partial{\Omega}$ be sufficiently smooth ...
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1answer
33 views

Second Existence theorem for Weak Solution in Evans

I have some questions about the proof of second existence theorem (Theorem 4 in 6.2) of weak solutions in Evans’ PDE: At the last part of step 4, it says we recall further from D.5 that the ...
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25 views

Bootstrap argument in De Giorgi's regularity theorem for elliptic PDEs

I was studying De Giorgi's regularity theorem for elliptic systems, and there is something I don't quite understand. The main part of the theorem is dedicated to the proof that certain functions (that ...
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1answer
29 views

Harmonic function on it's non-vanishing set

The problem I am stuck with is as follows: Let $\Omega\subset\mathbb{R}^n$ be a domain and let $u\in C^1(\Omega)\cap C^2(\Omega-\Sigma)$ be a non-negative function such that $\Delta u=0$ in $\Omega-...
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How can I calculate a normal vector of ellipsoid from using rotation matrix?

everyone. I'm doing calculations for a moving ellipsoid through differential equations. The point is that how I can get a normal vector of the ellipsoid on its surface point ($x_1,y_1,z_1$) with ...
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1answer
55 views

How can i prove that the elliptic intergal is equal to this double sum? $\Pi(n|m)=\int _0^{\frac{\pi}{2}}\frac{1}{(1-n\sin^2t)\sqrt{1-m\sin^2t}}dt=..$

How can i show that those are the same? $\Pi \left(n|m\right)=\int _0^{\frac{\pi }{2}}\frac{1}{\left(1-n\sin ^2t\right)\sqrt{1-m\sin ^2t}}dt=\frac{\pi }{2}\sum _{k=0}^{\infty }\sum _{j=0}^k\frac{\left(...
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2answers
68 views

Solution to the equation $\Delta f=\nabla f\cdot \nabla g$, where $g$ is radial.

I am looking for a non constant solution to the following equation $$ \Delta f-\nabla f\cdot\nabla g=0\qquad\text{on }\mathbb{R}^2, $$ where $g=g(r)$ is a positive, non constant function and $r=\sqrt{...
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14 views

Higher Sobolev regularity for Elliptic equation

Let $\Omega$ be a smooth subset of $R^d$. Suppose that $u\in W^{2,2}(\Omega)$ solves $$\text{trace}(AD^2u)=a_{ij}\partial_{ij}u=F,$$ in the weak sense, where $A=(a_{ij})_{ij}$ is a uniformly elliptic ...
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15 views

Lin second derivative estimates

I am in very trouble with how to proceed ( I don't need the exactly calculation) with these estimates. A Lin paper states the folliwing result: Consider a bounded, smooth domain $\Omega$ in $\mathbb{...
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27 views

Proof of mean value inequality

I was reading Elliptic PDE of second order by Gilberg. In that, I had a doubt in theorem 1 I do not understand how normal $\nu$ direction replaced by r which is again replaced by $\rho$ . Any Help ...
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1answer
28 views

Intuition from Boundary Point Lemma (Hopf Lemma)

Consider the classical Boundary Point Lemma: Let $L$ be an elliptic operator. Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
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Question on “without loss of generality” in an initial condition of a differential equation

We will also be interested in the next equation $$-cu'(x)-u''(x)=g(u(x)),\; x\in\mathbb{R},$$ $$u(-\infty)= l>\delta,\quad u(+\infty) = 0, $$ where $(c, u)\in\mathbb{R}_+\times C^\infty(\mathbb{R}...
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34 views

Weak Maximum Principle in nested domains for Elliptic PDEs.

I am having a bit of difficulty understanding the expositions in chapter 15 of the Gilbarg-Trudinger book. Consider an elliptic quasilinear PDE in an open connected domain: \begin{equation} \sum_{ij}...
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34 views

Aleksandrov maximum principle for semiconvex function

Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and convex function $w$ (it's equivalent saying that $u$ is semiconvex if exists ...
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9 views

What does the Linearised Version of a System of PDEs tell you about the Non-linear System?

I am trying to learn some basic PDE theory so apologies if this is a very naïve question. I am considering a paper by Sirakov on estimates for weakly coupled systems of elliptic PDE and it is stated ...
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22 views

Higher interior regularity for second order elliptic equation

Higher Interior regularity of second order elliptic equation i am not underderstand how the inequality (37) comes from (29)-(32) and (36), and how $\tilde{f}\in L^2(W)$ (?) $||\tilde{f}||_{L^2(W)}...
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27 views

Hessian Matrix Inequality

Consider the linear elliptic operator of the form $$Lu=a^{ij}D_{ij}u+b^iD_iu$$ with coefficient satisfying $$\lambda I \leq [a^{ij}] \leq \lambda^{-1} I , \,\,\,\,\,\,\,\,\,\,\,\,\,\, |b^i|\leq \mu$$ ...
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16 views

Weak variational formulation with non-homogeneous bondary condtitions

I am trying to find the weak variational form of the PDE \begin{align*} - \Delta u + u &= f(x,y), \hspace{.5cm} (x,y) \in T \\ u &= g_1(x), \hspace{.5cm} (x,y) \in T_1 \\ u &= g_2(y), \...
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1answer
29 views

The adjoint operator for the Elliptic equation

$L $ denotes the second order partial differential operator having either the form $$Lu=-\sum_{i,j} (a^{ij}u_{x_i})_{x_j}+ \sum _i b^iu_{x_i}+cu \tag{1}\label{1}$$ this is divergence form The ...
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83 views

Weak formulation and application of Lax-Milgram

I am working with the following PDE: \begin{align*} -\nabla \cdot (a(x) \nabla u) + b(x) u = f(x) &, \hspace{.5cm} x = (x_1,x_2) \in \Omega, \\ u = 0&, \hspace{.5cm} x \in \partial \Omega_1 ...
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1answer
156 views

Equivalence of two harmonic problems on different domains

I want to solve \begin{cases} \Delta u = 0,&\text{ in }\mathbb{R}^3\setminus B_1(0) \\ u=0,&\text{ as }\Vert x\Vert\rightarrow +\infty \\ u=1,&\text{ on }\partial B_1(0). \end{cases} I ...

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