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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3
votes
1answer
169 views

On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation $$\triangle \varphi = 4 \pi \rho \qquad (1)$$ defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and ...
4
votes
1answer
393 views

Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...
1
vote
0answers
36 views

Eigenvalue overdetermined problem

Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$: $$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\ u = 0 ...
2
votes
1answer
298 views

Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...
3
votes
1answer
315 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
1
vote
0answers
122 views

$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of $$u_{tt} + \Delta u =0$$ $$u|_{t=0}= u_0$$ $$u|_{t=T}=0$$ where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
2
votes
0answers
116 views

About a classic result from Han and Lin's PDE book

Let $A=a_{ij}$ be an $n \times n $ a symmetric matrix where the coeficients are in $L^{\infty}(B_r(0))$ and satisfies $$ \lambda |\xi|^2 \leq a_{ij}(x)\xi_i\xi_j \leq \alpha |\xi|^2, \ x \in ...
2
votes
1answer
244 views

Does this linear elliptic equation have a weak solution?

Let $Q = \Omega \times (0,C)$ where $\Omega$ is a bounded domain, write $(x,y) \in Q$ for $x \in \Omega$ and $y \in (0,C)$. Is the problem $$\Delta_{(x,y)}v = 0\quad\text{in $Q$}$$ $$\frac{\partial ...
3
votes
1answer
110 views

Laplacian on non-compact domains

I consider non-compact domains $\Omega$ with cylindrical ends. For example, $\Omega$ has a cover $\Omega_0 \cup (0,\infty) \times M$, where $\Omega_0$ has finite measure and $M$ is a compact manifold. ...
0
votes
1answer
123 views

Nontrivial solutions of a semilinear elliptic equation

What is known, for $N\geq3$, about the existence of nontrivial real-valued solutions $u=u(x)$ of the following semilinear elliptic equation: $$ \left\{ \enspace \begin{aligned} &\Delta u = f(u) ...
2
votes
2answers
191 views

“C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$”

Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation $$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim_{r ...
2
votes
0answers
54 views

Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
1
vote
0answers
132 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
-1
votes
1answer
160 views

Analysis of Sobolev spaces [closed]

I just wanted to know wthether the following is OK or not. Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
2
votes
0answers
169 views

Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
0
votes
0answers
125 views

Schauder estimate on a bounded domain

We know if $u:\mathbb{R}^{n}\to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by part, that $$ ||\Delta u||=||{\nabla}^2u|| $$ where $||\cdot||$ is the ...
1
vote
1answer
114 views

Uniqueness affine curvature

Let $\gamma_1,\gamma_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their (special)affine curvature, $\mu_1,\mu_2$ are equal, that is $\mu_1(\theta)=\mu_2(\theta)$, for ...
5
votes
1answer
180 views

How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$). Let $f\in ...
2
votes
0answers
37 views

Schauder-type estimates for polyharmonic operators in a smooth domain of $R^N$

Let $L$ be an elliptic operator of the form $$ Lu := (-1)^m \sum_{|\alpha|=2m} a_\alpha(x) D^\alpha u + \sum_{|\alpha|\leq 2m-1} b_\alpha (x) D^\alpha u $$ with smooth coefficients and $u$ defined ...
5
votes
3answers
368 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
2
votes
0answers
81 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
85 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
42 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
113 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 ...
7
votes
1answer
202 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 ...
2
votes
1answer
106 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
107 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
59 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 $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
1
vote
0answers
112 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, ...
12
votes
2answers
583 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
70 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
188 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
87 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
210 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
271 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
228 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
70 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 ...
1
vote
0answers
78 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
174 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
71 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
150 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 ...
7
votes
2answers
190 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 ...
13
votes
4answers
1k 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
70 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
175 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
150 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
234 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
2answers
141 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
150 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 ...
9
votes
1answer
261 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 ...