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
1
vote
0answers
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, $...
1
vote
0answers
24 views
How to proceed in this Boundary value problem where Eigen values are calculated numerically?
While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$)
\begin{eqnarray}
\lambda_h F''' - 2 \...
0
votes
0answers
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
votes
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 ...
-1
votes
0answers
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 $\...
1
vote
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 $\|\...
0
votes
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 ...
2
votes
0answers
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 $H^1(\Omega)$,...
2
votes
0answers
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
votes
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 $$\bigg(\frac{1}{h}∫_{-h}^...
5
votes
0answers
34 views
Convergence of free boundary minimal surfaces
I suspect the following statement is true:
Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and ...
2
votes
0answers
52 views
Analogous $H^1$-space for pseudo inner products
Perhaps this is a naive question but I could not find anything related to this.
Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...
1
vote
0answers
91 views
Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
7
votes
0answers
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 ...
1
vote
0answers
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
votes
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} = \...
0
votes
0answers
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{...
0
votes
0answers
39 views
Parabolic Brezis-Nirenberg problem
Has the parabolic version of the famous Brezis-Nirenberg problem ever been studied?
4
votes
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 \...
0
votes
0answers
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 ...
2
votes
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 ...
1
vote
0answers
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 ...
3
votes
0answers
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 ...
2
votes
0answers
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
votes
0answers
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?
$$
\int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
1
vote
0answers
63 views
Intuition from Hopf lemma (boundary point lemma )
Consider the classical boundary point lemma:
Let $L$ be an elliptic operator.
Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
3
votes
1answer
192 views
Aleksandrov maximum principle for semi-convex function
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
0
votes
0answers
41 views
How to find the PDE for the following transition density
Suppose I have the following two stochastic differential equations ($t\geq 0$)
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$
where $X = (X_t)$, $Z = (Z_t).$
Note that
$W=(...
2
votes
0answers
56 views
Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation
I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy
data for the Schrödinger equation if and only if
$$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
0
votes
0answers
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
vote
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:
$$
\begin{cases}
\frac{d^2V}{dx^2} = -\frac{q}{\epsilon}\left[p - n + \frac{N_0}{1+c_pp+c_nn}\right] \\\\
\frac{d}{dx}\left[\mu_nn\frac{dV}{dx} + D_n\frac{dn}{dx}\...
4
votes
1answer
120 views
References for systems of elliptic PDEs
I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...
2
votes
2answers
148 views
Representing a nonlinear elliptic PDE as an energy minimization problem
I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...
2
votes
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
$$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...
1
vote
0answers
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 $$\left\|e^{it\Delta}f\...
0
votes
0answers
80 views
integral of the laplacian to some power
I want to know the space of functions where the following quantity is uniformly bounded from above
$$\int_{K} (\Delta u)^j d\lambda< C,$$
where K is a compact and j is an integer number greater ...
9
votes
0answers
165 views
Non real eigenvalues for elliptic equations
I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
1
vote
0answers
73 views
Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE
In $\mathbb{R}^d$, consider the following equation
$$\Delta u -x\cdot \nabla u = f $$
where $f$ can be $C^\infty$ and decay like $e^{-\frac{c|x|^2}{2}}$.
I would like to know fundamental sol. to this ...
2
votes
0answers
63 views
Derivative estimates for Laplace eigenfunctions on Riemannian manifolds
In $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$), if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies $\Delta f+ f=0$, then we can write the following regularity/derivative estimates for $f$:...
4
votes
0answers
42 views
Dirichlet-to-Neumann map's estimate for mixed boundary value problems
The study on DtN or NtD maps for Dirichlet or Neumann boundary value problems (or PML for Helmholtz exterior problems) is pretty mature and there are tons of papers on this topic, yet I couldn't find ...
2
votes
0answers
157 views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
2
votes
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 ...
3
votes
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^...
2
votes
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 ...
2
votes
0answers
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 $F\in L^p(\Omega)$.
...
2
votes
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(\...
1
vote
0answers
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
votes
0answers
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
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
votes
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 ...
