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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2answers
158 views

vector valued pde's good reference

I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows. Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n ...
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1answer
197 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 ...
2
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1answer
174 views

Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$ Further assume a solvability condition $$\int_\Omega f ...
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2answers
97 views

Bound deg 3 partial differential operator on Laplace eigenfunction?

I am no expert on PDE and analysis but I am looking for certain technique from PDE. Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
7
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1answer
1k views

Yau's conjecture for positive Chern class

I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this ...
0
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1answer
108 views

analytic solution to elliptic PDE in R^n

I am looking for (minimal) conditions, which guarantee that the problem Lu = 0 in R^n, where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global ...
3
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0answers
130 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
1answer
250 views

A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation

While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two ...
2
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2answers
115 views

Is the left regularizer for elliptic BVP a left inverse for the principal part?

Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
3
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2answers
247 views

Non symmetric coefficient matrix for elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form $$ D_i(a_{i,j}D_ju)=0 \tag{1}$$ where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
2
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0answers
87 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 ...
5
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1answer
470 views

Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one. $I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...
3
votes
1answer
188 views

Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
2
votes
0answers
92 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 $f\in C^{0,\alpha}(B_1)$ ...
7
votes
2answers
238 views

Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
4
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1answer
162 views

Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
1
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1answer
211 views

Regularity of the right hand side (the source term) in Evans-Krylov theory

A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex ...
2
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1answer
128 views

The maximum in the Poisson problem on the cube with constant source

Question: Let us consider the Poisson problem on the square with constant source $1$ $$ \begin{cases} - \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\ u &= 0, \qquad \text{ on } \partial ...
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1answer
172 views

Mean value theorem for harmonic functions on ellipsoid

Is there any result like the mean value theorem for harmonic functions on ellipsoids (instead of sphere)?
0
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1answer
131 views

H_0^1 and C_infinity on the interior, does that imply classical limit is 0 on the boundary?

The solutions to the Dirichlet problem of elliptic PDE with smooth enough coefficients below to H_0^1 and also belong to C_infinity on the interior. Does that mean the classical limit of the function ...
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0answers
114 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 ...
14
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1answer
400 views

Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood

Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
1
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1answer
207 views

In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation $0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...
0
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1answer
127 views

Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following Theorem: ...
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2answers
535 views

Interior regularity for elliptic equations

The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for ...
3
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1answer
225 views

why is it called the diamagnetic inequality?

How is the Diamagnetic inequality born? Why is it call this name? Diamagnetic inequality: $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.
4
votes
1answer
512 views

Nash's paper on parabolic equations.

I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958). The author there establishes some a ...
1
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1answer
92 views

Boundedness of a given boundary value problem.

I've been given the following BVP: \begin{align*} -\Delta u = u- u^3,\: x\in \Omega \end{align*}\begin{align} u = 0,\: x\in \partial \Omega \end{align} where $\Omega\subset \mathbb{R}^N$ is bounded. ...
0
votes
1answer
186 views

Quantitative Global Schauder Estimates/Hölder Regularity

Consider the linear second order elliptic Dirichlet problem $$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$ Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda ...
3
votes
2answers
304 views

Schauder estimates for higher order linear elliptic operator on manifold

Hi! Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and ...
5
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0answers
610 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 ...
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0answers
110 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 ...
5
votes
3answers
357 views

Divergence form Elliptic PDE Removable Singularity/Regularity Question

Idea Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...
4
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2answers
223 views

T. Carleman's method on eigenvalues asymptotics

What is the best available less or more modern introduction to the subject?
2
votes
1answer
257 views

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
4
votes
1answer
170 views

Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...
1
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2answers
277 views

Variational problems whose lagrangian density depends on derivatives higher than 1.

The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
1
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1answer
303 views

Elliptic Differential Equations with rough boundary data

Question stated roughly: Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...
2
votes
1answer
189 views

Solvability of a nonlinear elliptic equation

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. ...
3
votes
1answer
271 views

Upper bounds for the solution of an elliptic PDE depending on a parameter.

Suppose I have the following PDE on $[0,1]^n$ $$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$ with periodic boundary conditions and $\int f(x) dx =0$ . ...
0
votes
1answer
129 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
1
vote
1answer
440 views

Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again. Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...
1
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1answer
217 views

Boundary Value Problem in the space of Distributions

I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the ...
3
votes
2answers
474 views

Elliptic regularity on bad domain

Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem $$ -\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega. $$ If $\varphi$ is a continuous function, ...
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0answers
125 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
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1answer
177 views

Reference to the Existence and Uniqueness of the PDE system

Hi all I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...
2
votes
1answer
213 views

Uniform upper estimate for derivatives of solution of an elliptic PDE

Let for every $i=1,\dots,n$ and $j=1,\dots,n$ be given functions $a^{ij}\colon\mathbb{R}^n\to \mathbb{R}$, $b^{i}\colon\mathbb{R}^n\to \mathbb{R}$, $c\colon\mathbb{R}^n\to \mathbb{R}$ from space ...
3
votes
1answer
335 views

Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let $\mathrm{id}$ be the identity operator, let $\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let $t > 0$ be a parameter. Does ...
1
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1answer
208 views

Harmonic/conformal map composition between manifolds in either order?

Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is ...
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0answers
667 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) ...