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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2
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0answers
43 views

probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions: Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...
1
vote
0answers
70 views

the most natural approach to solve an elliptic PDE in R^n [closed]

Let $n \geq 1, \lambda >0, f \in L^2(\mathbb{R}^n)$ and $b: \mathbb{R}^n \to \mathbb{R}^n$ a vector field satisfying $\mbox{div}(b) \in L^\infty(\mathbb{R}^n)$. What would be the most natural ...
0
votes
1answer
32 views

Oblique derivative smoothness of harmonic functions

Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
1
vote
0answers
38 views

Method of proving the regularity of the minimizer of geometric variational problems

Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer. We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants ...
6
votes
1answer
121 views

Mountain Pass theorem for minimization problems with constraints

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see ...
4
votes
0answers
187 views

Characterization of kernel of Bianchi operator

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the Frechet space of symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the Frechet manifold of metrics on ...
2
votes
0answers
61 views

Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful. I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...
2
votes
0answers
67 views

The level set of convolution

Suppose $u\in C^2(\bar \Omega)\cap C^\infty(\Omega)$ is a function which is compact supported on $\Omega$ with $u\equiv 0$ at $\partial\Omega$, and $\Omega\subset \mathbb R^3$. We also assume that ...
1
vote
0answers
41 views

Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$. Let $1<p<\infty$. Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
0
votes
0answers
65 views

Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$ \begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases} $$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...
0
votes
0answers
54 views

Linking theorem, elliptic pde

I am trying to solve some linear system of the form $$ \Delta \phi +p v(r)^{p-1} \psi = -f, \qquad \Delta \psi + qu(r)^{q-1} \phi =g $$ in $ \mathbb{R}^N$ where $ f,g$ are given and $ \phi,\psi$ are ...
12
votes
2answers
372 views

Does the Legendre-Hadamard condition imply a generalized Gårding inequality?

For simplicity, we restrict to constant coefficients. Let $A^{ij}_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition: $$ ...
0
votes
0answers
37 views

Regularity of solutions of strongly elliptic system: how smooth must the boundary be?

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. Let $A$ be a selfadjoint operator acting in $L^2(\Omega;\mathbb{C}^n)$. The operator $A$ is given by the ...
2
votes
1answer
119 views

Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...
4
votes
0answers
57 views

A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting: we have a non-negative supersolution $h$ of the following elliptic equation: \begin{equation} \Delta h + \|\nabla h\|^2 + f(x) \geq 0 ...
0
votes
1answer
125 views

Implicit function theorem for elliptic partial differential equations

Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...
5
votes
1answer
140 views

Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$

A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...
2
votes
0answers
212 views

A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
2
votes
0answers
44 views

Moser's iteration for non homogeneous quasilinear elliptic PDE

I want to know some reference of Moser's iteration for non homogeneous quasilinear uniformly elliptic PDE, in particular, $u_{ij}(\delta_{ij}+u_iu_j|\nabla u|^2)=0$. I want to know how to deal with ...
0
votes
0answers
76 views

Implicit function theorem on boundary points

I have the following examples: (1) $xy-1=0$ with $x\ge 0$. By the implicit function theorem, we can solve when $x\in(0, \infty)$. Here on the boundary we have $y=\frac{1}{x}\rightarrow \infty$ as ...
1
vote
0answers
53 views

Regularity of Schrödinger Resolvent

The following problem keeps bothering me: Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on ...
3
votes
1answer
143 views

$\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$. Does the following inequality (or something similar hold) for ...
0
votes
0answers
65 views

How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
3
votes
0answers
94 views

Continuously dependent on parameters [closed]

How do we check whether the solution is continuouly dependent on parameters? Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...
6
votes
2answers
141 views

Is there a maximum principle for stress in continuum mechanics?

I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the ...
1
vote
0answers
38 views

ABP estimates for semiconvex functions [closed]

Referring to the classical ABP Estimate (Gilbarg-Trudinger Lemma 9.2) I am looking for if such an estimate can be generalized to semiconvex functions. In an article of Trudinger (Comparison Principles ...
12
votes
4answers
593 views

Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
0
votes
1answer
50 views

Question regarding Laplace equation under Evans setting

All the following we use Evans notation. By Green's reconstruction formula, we could represent $u$ by $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy-\int_{\partial \Omega}u(y)\partial_\nu ...
2
votes
1answer
159 views

Eigenfunction on surface with boundary

Suppose we have a two-dimensional surface $M$ with smooth boundary $\partial M$. Equip $M$ with a metric $g$ such that the Gauss curvature $K$ of $M$ and geodesic curvature $\kappa$ of $\partial M$ ...
3
votes
0answers
67 views

Analytical solution of diffusion PDE with Robin boundary condition

I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk $\Omega$ with subject to Robin boundary conditions. The formulation is as ...
0
votes
1answer
139 views

Green's function and eigenvalues with multiplicity

Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
5
votes
1answer
113 views

significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...
1
vote
0answers
130 views

Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here. Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let ...
6
votes
0answers
150 views

What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?

We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$? The $p$-Laplacian ...
3
votes
1answer
170 views

Is this inequality true?

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...
0
votes
1answer
87 views

Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain. Definition of Newtonian ...
2
votes
0answers
79 views

Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
0
votes
0answers
81 views

Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...
1
vote
1answer
288 views

Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
1
vote
2answers
75 views

Solvability of quasilinear elliptic equations on closed manifolds

Is there any reference about solvability theory of quasilinear elliptic equations on closed manifolds? In particular, I am looking for solvability condition for function $f$ of following equation ...
4
votes
1answer
192 views

$C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular $C^0$ estimate for solutions of the Laplace equation with Neumann boundary conditions. More precisely, let $(M,g)$ be a ...
2
votes
1answer
146 views

Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
1
vote
0answers
141 views

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
6
votes
0answers
129 views

Local solvability of nonlinear elliptic boundary value problems

Malgrange proves the following statement regarding local solvability of (determined or underdetermined) nonlinear elliptic systems: Let $F_i(x,D^\alpha u)=0$ be a nonlinear elliptic system of order ...
1
vote
0answers
21 views

Differential operator with codimension 2 singularity in the domain

The soft version of the question is as follows: suppose I have a linear operator, and I know it is a 'nice' differential operator on its domain minus a singular set of codimension two. Does the ...
1
vote
1answer
80 views

Elliptic operator are unbounded [closed]

I am reading the book Index Theorem and the Heat Equation written by Peter.B.Gilkey. Here is my question: Let E be a hermitian vector bundle on a compact smooth manifold M. Let $D : ...
1
vote
1answer
146 views

2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...
4
votes
2answers
214 views

A question on certain elliptic PDE

Consider the elliptic PDE "CR" $$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$ And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ...
2
votes
1answer
145 views

Lp estimate for resolvent of Laplace operator

Consider for $1<p<\infty$ operator $A_p:L_p(0,1)\to L_p(0,1), \ D(A_p)=\{u\in W^2_p(0,1): u'(0)=u'(1)=0\}, \ A_pu=u''$, i.e. $L_p$-realisation of the Laplace operator with Neumann bcd on the ...
3
votes
1answer
76 views

Reference for Elliptic PDE on $\mathbb{R}^d$

Could anyone suggest a textbook, article, or lecture notes that covers elliptic PDE theory (existence, uniqueness, regularity) on all of $\mathbb{R}^d$, as opposed to the Dirichlet or Neumann problem ...