Questions tagged [elliptic-pde]

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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12 views

minimal assumption for elliptic equation

On the disc $\mathbb{D}$ on the disc with a metric $g=e^{2\lambda} \vert dz \vert^2$ and I consider either $$div_g(X)=e^{-2\lambda}div_e(e^{2\lambda} X)=f$$ where $div_e$ is the classical divergence, $...
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How to proceed in this Boundary value problem where Eigen values are calculated numerically?

While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$) \begin{eqnarray} \lambda_h F''' - 2 \...
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92 views

Schrodinger operator with matrix potential

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
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1answer
74 views

Extension of outer unit normal vector to interior

Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
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Understanding weak formulation of a linear elliptic pde [closed]

hey I'm working on weak formulation on an elliptic pde and I have these questions: Is there any difference between: $\Delta u=f$ in $\Omega$ and $\Delta u=f$ almost everywhere in $\Omega$ when $\...
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1answer
59 views

Poincaré-type Inequality

In Lieb's paper "On the lowest eigenvalue of the Laplacian for the intersection of two domains" one finds the following remark: Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...
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1answer
119 views

Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?

Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude ...
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26 views

Extension Sobolev functions across of lower dimensional subset

This question may be well-known to experts, but I am trying to get myself a rigorous proof. Consider open set $\Omega=B^n_1(0)\setminus B_1^k(0)$ in $\mathbb{R}^n$. If function $u$ is in $H^1(\Omega)$,...
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Is this $1$-form harmonic?

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...
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1answer
191 views

What is the motivation of the $L^p$ differentiability?

I was reading some papers and come up with the next definition : A function is differentiable in the $L^p$ sense at $x$ if there exists a real number $f'_p(x)$ such that $$\bigg(\frac{1}{h}∫_{-h}^...
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34 views

Convergence of free boundary minimal surfaces

I suspect the following statement is true: Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and ...
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52 views

Analogous $H^1$-space for pseudo inner products

Perhaps this is a naive question but I could not find anything related to this. Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...
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91 views

Spectrum of Laplacian-like operator

Let $\kappa_1, \kappa_2>0$ be fixed. Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by $$ A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
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149 views

Smoothness of solution map for PDE

I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
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32 views

Are there maximum principles related to the third boundary condition?

I am working on this problem \begin{equation} \begin{cases} u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\ u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0, \end{cases} \end{...
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1answer
137 views

Ancient Heat equation and Liouville's theorem

I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation: \begin{equation} u_{t} = \...
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34 views

First eigenfunction of the p-Laplacian in an interval $(a,b) \subset \mathbb R$

What is the explicit expression of the first eigenfunction $u$ of the $p$-Laplacian ($p>1$) in a bounded interval $(a,b) \subset \mathbb R$ (up to multiplicative constant)? \begin{equation} \begin{...
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39 views

Parabolic Brezis-Nirenberg problem

Has the parabolic version of the famous Brezis-Nirenberg problem ever been studied?
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1answer
213 views

A boundary Schauder estimate

According to Theorem 1.1' in this paper we have the following estimate on classical solutions $u \in C^2(\overline{B_1^+})$ of $-\Delta u = f \text{ in } B_1^+ = B_1 \cap \{x _n \ge 0 \}$ and $u = 0 \...
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51 views

Elliptic foliations of the plane

A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties: The differential operator ...
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0answers
79 views

Sufficient conditions for constant solutions

Let $u : \mathbb{R}^N \to \mathbb{R}$ be a smooth negative function that satisfies $$-f(x)u(x) + B(x) = \Delta u(x),~\forall x\in \mathbb{R}^N,$$ where $B(x)$ is a smooth positive function and $f$ is ...
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52 views

Normalized $p(x)-\mathrm{laplacian}$ is uniformly elliptic?

The normalized $p(x)-\mathrm{laplacian}$ is defined by $$-\Delta_{p(x)}^{N} u = -\operatorname{tr}\Big( \big( I + \frac{(p(x)-2)}{|Du|^{2}}Du \otimes Du\big)D^{2}u\Big)=0 ,$$ from now on, two ...
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Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates. In one of his papers, Lin proves the following result: Let's consider a ...
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40 views

Second derivative estimates for a subsolution of linear elliptic equation

Definition. 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 a convex function $w$. Note. Saying that $u$ is semiconvex is ...
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127 views

Integral estimate for the solution of the heat equation

Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality? $$ \int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
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63 views

Intuition from Hopf lemma (boundary point 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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1answer
192 views

Aleksandrov maximum principle for semi-convex function

Definition. 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 a convex function $w$. Note. Saying that $u$ is semiconvex is ...
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41 views

How to find the PDE for the following transition density

Suppose I have the following two stochastic differential equations ($t\geq 0$) $$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$ where $X = (X_t)$, $Z = (Z_t).$ Note that $W=(...
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56 views

Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation

I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy data for the Schrödinger equation if and only if $$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
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21 views

A little push in a test function

Let $M$ be an $n\times n$ matrix of real entries, which has at least one positive eigenvalue. Now consider $f$ a function of class $C ^{2}$ in a domain of $\mathbb{R} ^ n$, and $g$ a continuous ...
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1answer
101 views

How to solve numerically a system of 3 interdependent non-linear ordinary differential equations?

As per title, I need to solve this: $$ \begin{cases} \frac{d^2V}{dx^2} = -\frac{q}{\epsilon}\left[p - n + \frac{N_0}{1+c_pp+c_nn}\right] \\\\ \frac{d}{dx}\left[\mu_nn\frac{dV}{dx} + D_n\frac{dn}{dx}\...
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1answer
120 views

References for systems of elliptic PDEs

I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...
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2answers
148 views

Representing a nonlinear elliptic PDE as an energy minimization problem

I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation: $$\...
2
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1answer
120 views

Finite energy solution for Allen -Cahn equation

I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional $$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...
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43 views

On a interpolation inequality for the Schrödinger unitary group (NLS)

I'm trying to understand scattering for the classical nonlinear Schrödinger equation and for that i'm studying a scattering criterion on Tao's paper. At Lema 3.1 he states that $$\left\|e^{it\Delta}f\...
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80 views

integral of the laplacian to some power

I want to know the space of functions where the following quantity is uniformly bounded from above $$\int_{K} (\Delta u)^j d\lambda< C,$$ where K is a compact and j is an integer number greater ...
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165 views

Non real eigenvalues for elliptic equations

I am looking for an example of a pure second order uniformly elliptic operator $L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
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73 views

Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE

In $\mathbb{R}^d$, consider the following equation $$\Delta u -x\cdot \nabla u = f $$ where $f$ can be $C^\infty$ and decay like $e^{-\frac{c|x|^2}{2}}$. I would like to know fundamental sol. to this ...
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63 views

Derivative estimates for Laplace eigenfunctions on Riemannian manifolds

In $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$), if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies $\Delta f+ f=0$, then we can write the following regularity/derivative estimates for $f$:...
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0answers
42 views

Dirichlet-to-Neumann map's estimate for mixed boundary value problems

The study on DtN or NtD maps for Dirichlet or Neumann boundary value problems (or PML for Helmholtz exterior problems) is pretty mature and there are tons of papers on this topic, yet I couldn't find ...
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157 views

A basic question about the Spectral Theorem

Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e. $$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
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0answers
51 views

Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians

This is a follow-up question of this one. Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a ...
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1answer
118 views

Energy Decay of the functional $\int_{B_1} |Du|^2 +Au^2$

Suppose $u \in C^1(B_1)$ with $B_1 \subset \mathbb{R}^n$ such that $\Delta u =0$ weakly. We would have the energy decay estimate $$\int_{B_r} |Du|^2 \leq r^n \int_{B_1} |Du|^2.$$ Now suppose $u \in C^...
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0answers
64 views

Trace operators on submanifolds

In the following paper, Sobolev Spaces on Riemannian Manifolds with Bounded Geometry: General Coordinates and Traces https://arxiv.org/abs/1301.2539 The authors prove trace theorems for general ...
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63 views

$W^{2,p}$-estimates for Neumann boundary condition to Poisson equation

Consider the following Poisson-Neumann problem in a lipschitz bounded domain $\Omega\subset \mathbb{R}^3$: $-\Delta u=F,\quad \partial_n u\restriction_{\partial\Omega}=0$. Here $F\in L^p(\Omega)$. ...
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0answers
135 views

Solution to Heat Equation By Projection

Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation $$ \partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x), $$ for some fixed $p\in C^2(\...
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129 views

Is $(e^{u}+1)\Delta u+u=0$ the Euler-Lagrange equation of a functional energy?

Does there exist a functional energy $I$ such that $$(e^{u}+1)\Delta u+u=0$$ is the Euler-Lagrange equation associated with the energy functional $I$?
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36 views

A classic uniqueness problem in a constraint minimization problem

Consider the following constraint minimization problem $$ \inf_{\| u \|_p = 1} \int_{\mathbb{R}^N} |\nabla u|^2 + V(x)u^2 \,dx $$ where $\| \cdot \|_p$ is the $L^p$ norm, $2 < p < \frac{2N}{N-2}...
2
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1answer
166 views

Simple existence and uniqueness for second order and linear elliptic PDE

Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$. I am interested on the existence of solution for the following problem: given a continuous ...
2
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1answer
133 views

The Monge- Ampère equation with a non positive right hand side

Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the ...

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