All Questions
Tagged with ap.analysis-of-pdes calculus-of-variations
114 questions
3
votes
0
answers
72
views
Compactness of bounded index solutions of the Yamabe problem
Consider, a closed Riemannian manifold $ (M^n,g) $ , $ n \geq 3 $, with positive Yamabe invariant: $$ 0< Y(M, [g]):= \inf_{0<v \in H^1} Q_g(v), $$ where $$ Q_g(v) = \inf_{0 <v \in H^1} \...
- 457
5
votes
0
answers
608
views
What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?
1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
- 6,155
0
votes
0
answers
80
views
Relationship between two minimization problems
Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
2
votes
1
answer
141
views
Existence of first variation
I am trying to compute the first variation of the functional
$$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$
where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...
- 21
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
1
vote
0
answers
56
views
Finding thin plate spline subjected to boundary conditions
I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing.
This question is related to : Thin-Plate-Spline ...
- 115
1
vote
0
answers
38
views
Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
- 111
2
votes
0
answers
63
views
A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
- 809
3
votes
0
answers
84
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
1
vote
1
answer
98
views
How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let
$$
\begin{matrix}
F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\
(x,z,p) \mapsto F(...
- 217
2
votes
0
answers
77
views
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 ...
- 809
3
votes
0
answers
41
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 ...
- 471
6
votes
2
answers
2k
views
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
- 5,380
2
votes
0
answers
160
views
A naive question about the stable solution and Morse index of elliptic PDE
For example, for $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is stable if
$$
Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \...
- 809
2
votes
0
answers
109
views
relative entropy, Fisher information, and metric slope for non-convex domains
$\newcommand{\R}{\mathbb R}$ If $\Omega\subset \R^d$ is a convex domain it is well-known that the relative entropy
$$
\mathcal H(\rho)=
\int_{\Omega}\rho\log\rho \ \mathrm{d}x
\qquad \mbox{for }\rho=...
- 5,380
3
votes
1
answer
429
views
Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...
- 191
4
votes
1
answer
162
views
Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians
Consider a macroscopic free energy functional of the form
$$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
- 873
1
vote
0
answers
94
views
Nonlocal elliptic problem - what is its associated energy?
It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...
- 1,759
1
vote
1
answer
283
views
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}...
- 5,038
12
votes
0
answers
398
views
A model of pillows
(The same system with slightly different questions has been asked in MSE.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
- 134
2
votes
1
answer
307
views
Gradient descent relaxation dynamics of a Euler-Lagrange equation
I want to minimize the functional
$$
F=\int{L(u)}dx,
$$
where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...
- 159
4
votes
0
answers
97
views
Linking theorem
In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem:
$$
\begin{cases}
-\Delta u = \lambda u + |u|^{p-2}u, \Omega \\
u = 0, \...
5
votes
0
answers
206
views
Equality of weak solutions for inner products inducing equivalent norms
This is a repost of a now-deleted MSE question that did not get any comments or answers.
$\textbf{Background}$: This question is mainly about two basic questions: How do we systematically obtain the ...
- 153
1
vote
1
answer
158
views
How do I integrate this inequality that appears in a paper of Rabinowitz?
Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here.
I was reading a paper by Rabinowitz(this one to be more precise) and I came across ...
- 113
5
votes
1
answer
630
views
Uniqueness of Kantorovich potentials?
$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth.
Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
- 5,380
3
votes
0
answers
190
views
$C^1$-regularity of solution of a Dirichlet problem
I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
6
votes
1
answer
425
views
Lipschitz property of the symmetric rearrangement
I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...
- 457
0
votes
1
answer
413
views
What functions are equal to their symmetric decreasing rearrangement?
I am trying to understand the set
$$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^*(x)\}$$
where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if ...
- 537
3
votes
0
answers
65
views
Existence of ground state solutions for the critical exponent
I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.
The authors show that the above equation has a unique positive ...
- 537
9
votes
0
answers
289
views
Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?
$\newcommand{\id}{\operatorname{Id}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TM}{\operatorname{TM}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Cof}{\operatorname{Cof}}
\newcommand{\Det}{\...
- 6,741
2
votes
0
answers
86
views
Best constant for Sobolev-type inequality
I am currently reading a paper from Del Pino/Dolbeault about optimal constants of GNS inequalities (http://capde.cmm.uchile.cl/files/2015/06/pino2002.pdf).
The author wants to prove the following GNS ...
- 21
6
votes
1
answer
379
views
On the fundamental solution for elliptic PDE
In the well known paper by Littman-Weinberger-Stampacchia "Regular points for elliptic equations with discontinuous coefficients", the authors were able to prove the validity the following ...
- 253
1
vote
0
answers
85
views
Dirichlet-to-Neumann map for second order ODE
Problem statement
In a problem of interacting particles, I encountered a type of geodesic equation in $\mathbb{R}^n$ with an additional rotation and dilation term
$$
\ddot\gamma(t) + e^{t Q} \Lambda ...
- 1,081
2
votes
0
answers
64
views
A counterexample to regular boundary points for minimizers of variational integrals under subquadratic growth
Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\...
- 347
1
vote
1
answer
107
views
Characterization on smallest element in affine Sobolev subspace
Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that
$$
\phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$}
$$
(one can consider $C^{...
- 261
2
votes
1
answer
494
views
Fundamental Theorem of Gamma-Convergence
Let $(X, d)$ be a metric space and $F_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F_\varepsilon$ is an equicoercive sequence of functions on $X$, i.e. for all $t\in\mathbb{R}$ there exists ...
- 155
6
votes
1
answer
376
views
Lavrentiev phenomenon between $C^1$ and $C^2$
Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is
$$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
1
vote
0
answers
109
views
Is there a concentric map from the disk onto the ellipse with constant sum of singular values?
$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...
- 6,741
3
votes
2
answers
364
views
Gradient estimates for a boundary value problem
$\newcommand{\avint}{⨍}$
Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE:
$$
\begin{cases}
-\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\
w=0 \...
- 261
5
votes
2
answers
746
views
Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior?
Suppose $\Omega$ is a bounded domain in $\mathbb R^3$ with Lipchitz boundary $\partial\Omega$, and $u\in H_0^1(\Omega)\cap C(\Omega)$. Is $u$ continuous to the boundary i.e. do we have $u \in C( \...
- 401
0
votes
0
answers
142
views
Variational formulation for an elliptic boundary value problem
I'm trying to determine the variational formulation of
$$
\begin{cases}
-\Delta u(\mathbf{x})=1, & \mathbf{x}\in (0,1)\times (0,1) \\
-\partial_{x}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=0, & ...
user165558
4
votes
2
answers
226
views
Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight
Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
- 1,101
2
votes
0
answers
84
views
A problem of uniqueness
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem:
$$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$
$$ u(x,0)=f(x),\quad\forall x\...
- 339
3
votes
1
answer
137
views
Variational formulation of an elliptic pde
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, what is the variational formulation of the following problem:
$$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$
$$ V(x,...
- 339
4
votes
0
answers
135
views
Determination of the nature of stationary values in variational calculus
In variational calculus, when we solve the Euler-Lagrange equation $\frac{d}{dx}L_p(u',u,x)-L_z(u',u,x)$, where $L=L(p,z,x)$, to find stationary inputs of the functional
$$
I[u]=\int_0^1 L(u',u,x)dx,
$...
- 1,755
1
vote
0
answers
102
views
Dislocations and Random Matrix Theory
Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help.
By ...
- 595
2
votes
0
answers
196
views
Have you seen this PDE before?
Consider the second-order nonlinear PDE
$$(\partial_x\partial_y\varphi)\cdot\varphi = \partial_x\varphi\,\partial_y\varphi.$$
This PDE is solved by all ('separable') functions $\varphi\in C^2(\Omega)$ ...
- 463
1
vote
0
answers
126
views
Does my functional satisfy the Palais Smale condition?
Consider the functional
$$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$
where $u:\mathbb{R}^2 ...
- 383
3
votes
1
answer
138
views
Boundary regularity of rectifiable multiplicity 1 hypercurrents
Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
- 31
2
votes
0
answers
286
views
Modified variational formulation of heat equation
The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to
$$
u_t = \Delta u,
$$
subject to certain boundary conditions and can alternatively be described, in variational form, as ...
- 5,405