# 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.

669
questions

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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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24 views

### 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 ...

**2**

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**1**answer

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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39 views

### 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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**1**answer

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 $\|\...

**0**

votes

**1**answer

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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97 views

### 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 \...

**2**

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**1**answer

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}^...

**5**

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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 ...

**2**

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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} \...

**7**

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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{...

**3**

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**1**answer

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?

**4**

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**1**answer

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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**0**answers

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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96 views

### 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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**0**answers

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 ...

**2**

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**0**answers

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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**0**answers

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 $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...

**3**

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**1**answer

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=(...

**2**

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**0**answers

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}\...

**0**

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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 ...

**1**

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**1**answer

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}\...

**4**

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**1**answer

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'...

**2**

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**2**answers

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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**1**answer

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
$$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...

**1**

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**0**answers

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\...

**0**

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**0**answers

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 ...

**9**

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**0**answers

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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**0**answers

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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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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**0**answers

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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**0**answers

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 ...

**3**

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**1**answer

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^...

**2**

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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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**0**answers

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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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(\...

**1**

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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 $I$?

**0**

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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**

votes

**1**answer

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**

votes

**1**answer

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 ...