Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

learn more… | top users | synonyms

3
votes
1answer
147 views

Green's function for *GJMS* operator

Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
2
votes
1answer
171 views

A proof of energy functional appearing in the regularity of elliptic and parabolic equations

I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...
-1
votes
1answer
77 views

Regularity of solutions for a non linear elliptic equation

Let $v_k$ be a radial sequence of function that satisfies in $\Omega\subset\mathbb{R}^4$ $(-\Delta)^2 v_k=e^{v_k}$ $v_k(x)\leq v_k(0)=0$ $\left\Vert (-\Delta)v_k\right\Vert_{L^1(B_R(0))}=O(1)\qquad ...
5
votes
0answers
101 views

Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
5
votes
0answers
578 views

elliptic regularity on manifolds

Hello! I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic ...
4
votes
0answers
93 views

Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
4
votes
0answers
256 views

Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
4
votes
0answers
308 views

Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...
3
votes
0answers
41 views

Hopf Lemma for strong solutions

The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and ...
3
votes
0answers
96 views

Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition. What can ...
3
votes
0answers
328 views

problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution? $$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...
2
votes
0answers
29 views

reference on existence result for nonlinear elliptic PDE

During my work, I came to the question of existence of weak solutions to the following elliptic equation $$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$ with ...
2
votes
0answers
80 views

slightly subcritical elliptic pde; the linearized equations

Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems $$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...
2
votes
0answers
70 views

Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain

Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
2
votes
0answers
69 views

A question about fractional integrals

I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ...
2
votes
0answers
84 views

A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...
2
votes
0answers
91 views

Regularity of solution of nonlinear equation

Hi! Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball $B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$. Let $f\in C^{0,\alpha}(B_1)$ ...
2
votes
0answers
178 views

Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of $$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$. I also assume that ...
2
votes
0answers
160 views

A first order ODE with Sobolev estimates

Suppose $f: {\mathbb R}\to {\mathbb R}$ is a continuous function satisfying $$f(s) = c_- >0,\forall s<-T; f(s) = c_+<0, \forall s>T.$$ (For simplicity we may even assume that $f$ is ...
2
votes
0answers
249 views

A free boundary problem by finite difference method

I wanna discretize the following free boundary problem Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$. I apply finite ...
2
votes
0answers
247 views

Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?

Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to ...
2
votes
0answers
277 views

Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations: $-\Delta\phi+a(x)\phi=b(x)$ where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...
2
votes
0answers
262 views

Poisson problem with a “scaled” Laplacian.

Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form $$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...
2
votes
0answers
214 views

Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer. Background: When ...
1
vote
0answers
19 views

Constructing a family of domains for application of method of continuity in optimal transportation

Anyway can help me about this paper? http://arxiv.org/pdf/math/0601086v4.pdf I want to ask page 20, The author want to construct a family of subdomain for using method of continuity. But I can't ...
1
vote
0answers
140 views

Bunimovich stadium bouncing ball

http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarring-for-the-bunimovich-stadium/ I cannot relate how the eigenfunctions normalizaed correspond to the probabality distribution of the ...
1
vote
0answers
104 views

Result like Brezis-Kato Lemma for Biharmonic equation.

Is there any result for the 4th order elliptic pde,like we do have for 2nd order pde: Consider the equation,\ $\Delta^2 u = a(x)u$ and $a(x)\in L^{\frac{n}{4}}(\Omega)$, where $\Omega$ is bounded. If ...
1
vote
0answers
121 views

Compactness of solutions of elliptic equation

Consider the following nonlinear elliptic equation $$ -\triangle u + u + u^3 = g, \quad x \in R^3. $$ If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...
1
vote
0answers
149 views

On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest eigenvalue $\mu_{3}$ for the differential equation $\Delta ...
1
vote
0answers
190 views

References? Stability of the Cauchy problem for elliptic and backward-parabolic operators.

Dear community. I'm looking for survey articles about recent and old results about stability for the elliptic and backward-parabolic Cauchy problem. For example: Take $\partial_t u + ...
0
votes
0answers
34 views

Integrability of $D^2u$ for $\infty$-harmonic function $u$?

Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, ...
0
votes
0answers
53 views

Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...
0
votes
0answers
72 views

Wave operator for focusing NLS

Consider the NLS equation \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u+u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\right. \end{equation} where ...
0
votes
0answers
50 views

Mean value formula for high order elliptic operators

Weyl's lemma says any distributionally harmonic function $u\in L^1_{\text{loc}}$ is harmonic. One way to prove it is as follows. Take radially symmetric compactly supported and smooth approximation ...
0
votes
0answers
35 views

basic doubt in a free boundary problem for the p-Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf In this article the author consider $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...
0
votes
0answers
21 views

Vanishing of non commutative ( Wodzicki) residue on pseudo differential projections

Its a known fact that the non-commutative (Wodzicki) residue of a pseudo-differential projection is always zero. My question is: Is it possible to get this result by looking at structure of the ...
0
votes
0answers
59 views

A density result for biharmonic functions

Let $U_1, U_2\subset\mathbb R^3$ open and homeomorphic to the open unit ball, with sufficiently smooth boundary, such that $\overline{U}_1\subset U_2$. Is is true that every biharmonic function $u$ ...
0
votes
0answers
220 views

Analytic solution of Poisson's equation

Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square. Impose a von Neumann boundary condition ...
0
votes
0answers
107 views

second derivatives of initial conditions for a non-linear PDE

Hello, I have the following problem: Having a non-linear PDE of second order, there are the initial conditions $F(x,0)=g(x)$, $\partial_yF(x,0)=h(x)$. The question is: Can the second derivatives ...
0
votes
0answers
600 views

Explicit analytic solution of an 2D Poisson equation

I am not very familiar with analytic solution of PDEs. Here is the problem I don't know how to solve: Let $\Omega$ be the unit square $(0,1)^2$, we consider the elliptic equation $-div(k(x,y) ...