All Questions
Tagged with ap.analysis-of-pdes calculus-of-variations
66 questions with no upvoted or accepted answers
22
votes
0
answers
2k
views
Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
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- 6,741
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
9
votes
0
answers
289
views
Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?
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- 6,741
8
votes
0
answers
295
views
What is nice in gradient flows?
First of all, I am sorry for such naivety. I have not all the intuition in hard analysis as I wish.
I am studying Perelman's work and his big first contribution is to prove that the Ricci flow is in ...
- 1,774
7
votes
0
answers
619
views
Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 1,309
7
votes
0
answers
343
views
Methods of variational calculus in analytic number theory
What methods of calculus of variations have been used in analytic number theory?
I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...
- 1,594
5
votes
0
answers
608
views
What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?
1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
- 6,155
5
votes
0
answers
206
views
Equality of weak solutions for inner products inducing equivalent norms
This is a repost of a now-deleted MSE question that did not get any comments or answers.
$\textbf{Background}$: This question is mainly about two basic questions: How do we systematically obtain the ...
- 153
5
votes
0
answers
101
views
Minimisers and critical points of variational integrals
In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral:
\begin{equation*}
I[u]=\int_{\...
- 347
5
votes
0
answers
487
views
"Euler system" in Christodoulou's The Action Principle and PDEs
In The Action Principle and PDEs Christodoulou spends some time describing what he calls the Euler system associated to a system of variational PDEs (sections 2.5-7, 6.2). Briefly, given a bundle $E\...
- 21.5k
4
votes
0
answers
97
views
Linking theorem
In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem:
$$
\begin{cases}
-\Delta u = \lambda u + |u|^{p-2}u, \Omega \\
u = 0, \...
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
$$
I[u]=\int_0^1 L(u',u,x)dx,
$...
- 1,755
4
votes
0
answers
254
views
Lower semi-continuity of integration
I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
- 5,405
4
votes
0
answers
230
views
Sobolev spaces of maps between manifolds and the Palais-Smale Condition
I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
- 139
3
votes
0
answers
72
views
Compactness of bounded index solutions of the Yamabe problem
Consider, a closed Riemannian manifold $ (M^n,g) $ , $ n \geq 3 $, with positive Yamabe invariant: $$ 0< Y(M, [g]):= \inf_{0<v \in H^1} Q_g(v), $$ where $$ Q_g(v) = \inf_{0 <v \in H^1} \...
- 457
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 $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
3
votes
0
answers
84
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
3
votes
0
answers
41
views
Functional of fully nonlinear equations
Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
- 471
3
votes
0
answers
190
views
$C^1$-regularity of solution of a Dirichlet problem
I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
3
votes
0
answers
65
views
Existence of ground state solutions for the critical exponent
I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.
The authors show that the above equation has a unique positive ...
- 537
3
votes
0
answers
183
views
Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?
Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$u(t) \leq \psi(t) ...
- 73
3
votes
0
answers
119
views
Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS
Consider the mass-critical defocusing NLS in dimension $d\geq 1$:
$$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$
Define the mass $M(u)$ and scattering size $S(u)$ of the ...
- 873
3
votes
0
answers
127
views
Existence of at least one positive solution for semilinear biharmonic equation with critical exponent
Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{...
- 371
3
votes
0
answers
105
views
Can Mumford-Shah functional be adapted to lower $L^1$ space?
The well know Mumford-Shah functional functional
$$
F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1
$$
where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
- 679
3
votes
0
answers
146
views
Variational Principle for a System of Differential Equations
I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v \...
- 516
2
votes
0
answers
63
views
A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
- 809
2
votes
0
answers
77
views
Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem
Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
- 809
2
votes
0
answers
160
views
A naive question about the stable solution and Morse index of elliptic PDE
For example, for $$
-\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
$$
\mathcal H(\rho)=
\int_{\Omega}\rho\log\rho \ \mathrm{d}x
\qquad \mbox{for }\rho=...
- 5,380
2
votes
0
answers
86
views
Best constant for Sobolev-type inequality
I am currently reading a paper from Del Pino/Dolbeault about optimal constants of GNS inequalities (http://capde.cmm.uchile.cl/files/2015/06/pino2002.pdf).
The author wants to prove the following GNS ...
- 21
2
votes
0
answers
64
views
A counterexample to regular boundary points for minimizers of variational integrals under subquadratic growth
Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\...
- 347
2
votes
0
answers
84
views
A problem of uniqueness
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem:
$$\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
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
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
2
votes
0
answers
43
views
Commonly used metrics to compare two Young measures
Let $\Omega\subset \mathbb{R}^n$ be a bounded open set, $K\subset \mathbb{R}^d$ be a compact set, and $M_1(K)$ be the set of probability measures on $K$. Then a Young measure is defined as a $\textrm{...
- 503
2
votes
0
answers
105
views
Energy functional continuous with respect of time $t$
I am studying a paper of Liu Yacheng which named "On potential wells and applications to semilinear hyperbolic equations and parabolic equations" it considers a nonlinear parabolic equation
\begin{...
- 287
2
votes
0
answers
162
views
Pohozaev identity and related non-existence result for a nonlinear problem
Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional
$$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...
user124345
2
votes
0
answers
113
views
Laplacian variational problem with asymptotically quadratic term
Consider the functional
$$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$
where $\Omega$ is a bounded smooth domain.
The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...
- 839
2
votes
0
answers
214
views
Variational formulation for elliptic interface problem
Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...
user60665
2
votes
0
answers
140
views
Question on the differentiability of the solution mapping in the obstacle problem
I'm looking for a reference for the following. Take the obstacle problem:
$$\int_\Omega \nabla u \nabla (v-u) \leq \int_\Omega f(v-u)$$
for a function $u \in K:=\{v \in H^1_0(\Omega) : v \geq \varphi\}...
- 73
2
votes
0
answers
64
views
The continuity of $L^2$ gradient on moving domain
I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(...
- 679
2
votes
0
answers
146
views
Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$
Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\}
...
- 5,380
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
$$\boldsymbol{\...
- 111
1
vote
0
answers
94
views
Nonlocal elliptic problem - what is its associated energy?
It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...
- 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
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
$$
\ddot\gamma(t) + e^{t Q} \Lambda ...
- 1,081
1
vote
0
answers
109
views
Is there a concentric map from the disk onto the ellipse with constant sum of singular values?
$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...
- 6,741
1
vote
0
answers
102
views
Dislocations and Random Matrix Theory
Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help.
By ...
- 595
1
vote
0
answers
126
views
Does my functional satisfy the Palais Smale condition?
Consider the functional
$$ 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