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.
1,100
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On the weak derivative of $|u|^{(p-2)/2}u$
Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$.
How can I show that
$$
D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1}
$$
or how can I show that, ...
- 143
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1
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Derive elliptic maximum principle from weak derivatives
Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that
$$...
- 2,710
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Well posedness of the Plateau problem under lack of uniqueness
The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not.
Premises
I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...
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0
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Regularity of nonlinear equation for fractional laplacian
For a bounded smooth domain $\Omega \subset \mathbb{R}^n$, suppose $u\in H_0^1(\Omega)$ is a solution to
$$-\Delta u= |u|^{\frac{4}{n-2}}u+u \text{ in } \Omega.$$
one can employ Moser iteration to ...
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0
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24
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A question to the proof of Lemma 9 in "Multiple solutions for the Brezis-Nirenberg problem"
I'm currently puzzled by the final portion of the proof for Lemma 9 in the paper "Multiple solutions for the Brezis-Nirenberg problem"(DOI:10.57262/ade/1355867873). In particular, I'm unsure ...
- 631
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0
answers
60
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Techniques to estimate PDE which are elliptic in some directions and degenerate in others
I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
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1
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Elliptic regularity when the Lagrangian is possibly infinite
I want to solve variational problems of the form
$$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$
where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
CommunityBot
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Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
- 595
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When considering an equation on flat torus, can we treat it as an equation on a square with opposite sites topologically identified? [migrated]
I need to use some theorems (which is originally for $2$-dim bounded domain in Euclidean space) on torus, what should I do to the equation? Can I just simply treat it as an equation on a square with ...
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Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...
- 505
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0
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views
Elliptic regularity for Dirichlet problem
Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective ...
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votes
1
answer
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Uniqueness of constructed solutions to the Helmholtz equation
My question is regarding the inhomogeneous Helmholtz equation on $\mathbb{R}^3$ with real wavenumber $k$ and outgoing radiation condition
\begin{equation}
\Delta u + k^2 u = - f \quad \text{and} \quad ...
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Any theory on the elliptic operator $Lu=\Delta u + b_iu_i + cu$ when $c>0$
I wonder if there are theories on elliptic operator $$Lu=\Delta u + b_iu_i + cu$$ when $c>0$, when $c<0$, we are glad to have maximum principle, so the bijectivity can be easily analyzed, but I ...
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To study the elliptic PDE on complex manifold, when can we treat it as the real case?
I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying
$$\Delta_c u = f(x,u),$...
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votes
1
answer
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Can gradient zero implies that a function is constant with Hörmander vector fields
Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
CommunityBot
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A question about regularity results in the Elliptic case which are given by Schauder theory
I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
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2
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Local existence of non-trivial solutions to first-order linear elliptic system of PDE
This question came up when I was trying to find out the details about the existence of isothermal coordinates for surfaces.
Given a surface in $\mathbb{R}^3$, at least $C^2$ for simplicity, at any ...
CommunityBot
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2
votes
0
answers
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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
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About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
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1
vote
1
answer
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Integral identity for critical points of the Ginzburg-Landau functional
I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional
$E_\epsilon(v) = \frac{1}{2}...
CommunityBot
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votes
2
answers
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views
Nontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by
$$ v(r,\theta) = u(\frac{1}{r},\theta)$$
is also harmonic for $r>0$. Note that the Kelvin ...
- 4,089
1
vote
1
answer
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views
How to solve numerically a system of 3 interdependent non-linear ordinary differential equations?
As per title, I need to solve this:
$$
\begin{cases}
\frac{d^2V}{dx^2} = -\frac{q}{\epsilon}\left[p - n + \frac{N_0}{1+c_pp+c_nn}\right] \\\\
\frac{d}{dx}\left[\mu_nn\frac{dV}{dx} + D_n\frac{dn}{dx}\...
CommunityBot
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votes
0
answers
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About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
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votes
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answer
242
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Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
- 37.6k
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votes
2
answers
362
views
Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
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votes
0
answers
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Visualization of an oscillation lemma
How can one visualize Theorem 4.2 on page 31 of this paper by Seregin, Silvestre, Šverák and Zlatoš?
On the other hand, I have a clear visualization of a related result about how oscillation decay ...
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answers
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Gradient estimates of linear elliptic PDE
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
$$u(x)\Big|_{\...
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votes
1
answer
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Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood?
Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
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1
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Application of Yamabe and Liouville type equation
Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
The Yamabe Type Equation (for $n>2$):
\begin{equation}
-\...
CommunityBot
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views
Elliptic PDEs in BSDEs and in optimal control
This soft/reference question is related to this MO post of a similar nature.
What are some examples of elliptic PDEs appearing in control and BSDEs?
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votes
1
answer
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Regularity of solutions to $-\Delta u = \operatorname{div} F$, $F\in L^1$
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.
What are the regularity results for solutions to
$$
-\Delta u= \operatorname{div} F,
\qquad
F\in L^1(\Omega,\mathbb{R}^n)?
$$...
CommunityBot
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votes
0
answers
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Question about the formula of Green function of Laplacian on sphere
I'm reading a paper which said that
the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
- 595
1
vote
1
answer
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views
What can one say about the Dirichlet problem for Schrödinger equation with negative potential?
Consider the Schrödinger type equation in $\Bbb R^2$:
$$
\Delta f(x,y)+c(x,y)f(x,y)=0
$$
where $c(x,y)$ is a positive (!) function everywhere analytic on the plane, and $\Delta$ is the Laplace ...
CommunityBot
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1
answer
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Derivative of Moreau envelope in Hilbert space with respect to regularization parameter $\lambda$ using Hopf-Lax formula?
Let $\mathcal H$ be a Hilbert space, $f \colon \mathcal H \to (- \infty, \infty]$ a proper, convex, lower semi-continuous function and $\lambda > 0$.
The $\lambda$-Moreau envelope of $f$ is
$$
f_{\...
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0
answers
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Some questions about the concept of stable solution of elliptic PDE
For $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is stable if
$$
Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall ...
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votes
2
answers
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Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...
- 20.7k
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0
answers
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Proof that, for $u \in H^1$, $\{ u > \alpha \}$ is a quasi open set
I am reading the monograph by A. Henrot, Extremum problems for eigenvalues of elliptic operators. In chapter 2, the notion of a quasi-open set is defined (see the relevant definitions at the end of ...
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votes
1
answer
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Solutions to $\Delta u\ge u^2$
Let $(M,g)$ be a complete Riemannian manifold. Suppose that $u$ is a nonnegative solution to $\Delta_gu\ge u^2$. Does it follow that $u$ must be identically 0?
I know that the answer to above ...
CommunityBot
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2
votes
1
answer
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Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
CommunityBot
- 1
1
vote
0
answers
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Regularity of Feynman-Kac formula for a simple diffusion
Let consider the diffusion process given by:
$$dX_t = \alpha(X_t) dW_t$$
where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
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votes
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Examples of symmetry-breaking solitons which retain a subgroup symmetry
There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions.
However, all symmetry breaking soliton examples I have seen go from the ...
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votes
1
answer
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Elliptic PDEs in Finance
In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
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0
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Any solution of an evolution problem tends to a steady state in $L^2$?
I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
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votes
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answers
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$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature
This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ .
The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
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votes
1
answer
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views
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
- 4,734
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votes
0
answers
48
views
Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 5,632
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votes
0
answers
66
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Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem
Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
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2
votes
0
answers
38
views
Functional of fully nonlinear equations
Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
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2
votes
1
answer
64
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Existence of solution to Cauchy boundary value problem in Lipschitz class of functions
For a research question I have run into the following problem that seems intuitively true but I do not know how to prove it and am not sure in which generality.
Let $\Omega\subset \mathbb{R}^2$ be a ...
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0
answers
133
views
Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
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