Questions tagged [calculus-of-variations]

Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.

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2
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0answers
35 views

Prove the equicontinuity of a maximizing sequence

Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...
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65 views

Area of Disc that Intersects Another under Smooth Flow

The following question can be asked in any $\mathbb{R}^n$ for $n > 1$, but the case of interest is (thankfully) the case $n = 2$. The formulation of the problem with discs isn't actually critical ...
3
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1answer
111 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^...
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31 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}...
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25 views

Controlling oscillation of a p-harmonic function in a small ball

Given $\Omega\subset \mathbb R^N$ open. And $u:\Omega\rightarrow \mathbb R$ be a $p$-harmonic function. That is it minimizes the functional: $$ \min _{v\in W_{\varphi}^{1,p}(\Omega)}\int_{\Omega}|\...
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0answers
66 views

Help showing F is weakly lower semicontinuous

Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...
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24 views

Given a sequence of self-adjoint operator $L_w$, find the parameter $w$ maximizing the spectral gap of $L_w$

Let $(E,\mathcal E,\mu)$ be a probability space, $$L^2_0(\mu):=\left\{g\in L^2(\mu):\int g\:{\rm d}\mu=0\right\}$$ and $k\in\mathbb N$. Suppose for each $w\in L^2(\mu;\mathbb R^k)$, there is a ...
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0answers
36 views

interchange of minimization and integration on a product space

Let $(\Omega,\mathcal A,\mu)$ be a measure space $E$ be a $\mathbb R$-Banach space $K$ be a closed subspace of $E$ $\Gamma:=\left\{x:\Omega\to E\mid x\text{ is }\mathcal A\text{-measurable and }x(\...
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37 views

Optimization of an integral functional when the multiplier rule yields no useful information

Let $(E,\mathcal E,\lambda)$ be a measure space $\mu\ll\lambda$ be a probability measure on $(E,\mathcal E)$ $p\in[1,\infty)$ $k\in\mathbb N$ $f:E\times\mathbb R^k\times\mathbb R^k\to[0,\infty)$ such ...
2
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0answers
65 views

Link between Yamabe invariant and Yamabe equation

I am trying to understand the solution to the Yamabe problem as presented by Lee and Parker. It seems to me that the constant $\lambda$ which appears in the Yamabe equation $$\square\varphi = \lambda \...
1
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0answers
55 views

Can we reduce the maximization of this integral to the maximization of the integrand?

I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
7
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2answers
291 views

Do Minkowski sums have anything like calculus?

Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums? For example, it is known ...
1
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1answer
132 views

Converting an integral equation into a differential equation

Let $a, b \in \mathbb R^n$ and $f, g \in L^1 [0,1]$. Assume for all $h \in AC[0,1 ]$ (space of absolutely continuous functions) following integral equality holds $$ \int_{0}^{1} \langle f(t) , h(t) \...
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0answers
64 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{...
2
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1answer
137 views

Properties of the topology of sequential convergence $\tau_{seq}$

Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_{seq}$ has the following ...
3
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0answers
193 views

Maximize an $L^p$-functional subject to a set of constraints

Let $(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces $f\in L^2(\lambda)$ $I$ be a finite nonempty set $\varphi_i:E'\to E$ be bijective $(\mathcal E',\mathcal E)$-measurable ...
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0answers
51 views

When is $ \int_{0}^{1} F (t , x(t) , x' (t)) \; dt$ finite for all trajectories $x(t)$?

Let $f : AC[0, 1] \to [- \infty , + \infty]$ be defined by $ f(x(.)) := \int_{0}^{1} F (t , x(t) , x' (t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}...
2
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1answer
166 views

Is $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$ differentiable?

Let $f : AC[0, 1] \to R$ be defined by $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}$, and $F: R^n \to R$ is ...
2
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0answers
154 views

Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?

Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
1
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0answers
29 views

Variational inference: Does the natural gradient follow (Fisher-Rao) geodesics locally?

Amari's natural gradient descent is a well-known optimisation algorithm for functionals defined on statistical manifolds. It consists of preconditioning the vanilla gradient descent update rule with ...
2
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0answers
33 views

Extremal field implies minimality — low regularity assumptions?

In A stability theorem for minimal foliations on a torus, Moser is studying variational integrals of the following form: \begin{equation*} \mathcal{F}(u) = \int F(x,u(x),Du(x)) \, dx \end{equation*} ...
1
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0answers
99 views

Gradient of squared riemannian distance on complete manifold

Let $\theta: M \times M \to \mathbb{R}$ the squared distance function $\theta(x,y)=d(x,y)^{2}$ on complete Riemannian manifold $M$. I would like to calcule the gradient of $d^{2}$, where $d^{2}_{y}(x)=...
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0answers
135 views

Minimising an Integrated Relative Entropy Functional

Suppose I am given A probability distribution on $\mathbf R^d$, with density $\pi (x)$. A family of transition kernels $\{ q^0 (x \to \cdot) \}_{x \in \mathbf R^d}$ on $\mathbf R^d$, with densities $...
1
vote
1answer
153 views

Construction of elliptic equation with Neumann boundary condition from a minimization problem

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem. Let $B=B_1 \subset \mathbb{R}^3$ and $E : H^1(B) \to \mathbb{R}$ $$E(...
0
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1answer
119 views

Supremum of log(E[X]]-E[log(X)]

I've been tackling some questions on probability theory and got stuck on this one. Determine $$\sup_{1≤X≤b} \ \log(E[X])-E[\log(X)]$$ where $X$ is a random variable defined in $[1,b]$. In other words,...
1
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0answers
57 views

Global solution of nonlinear Schrödinger equation via blow-up argument

Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$. I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$, then there is ...
4
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1answer
144 views

Minimizing sequence $\implies$ Palais–Smale sequence

Set $F:\mathbb{R}^n\rightarrow \mathbb{R}$ a $C^2$-function that is bounded from below. Set $x_n$ a minimizing sequence, i.e., $F(x_n)\to \alpha = \inf F$. I want to prove that under the assumption of ...
3
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0answers
93 views

General way to find Nash equilibrium in continuous game

I'm really interesting how to find Nash equilibrium in a continuous game with two players in the general case. Let's consider a game with continuous utility functions $F_1, F_2 : [0, 1] \times [0, 1]...
2
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1answer
105 views

Generalization of du Bois-Reymond Lemma into dimension 2?

The du Bois-Reymond lemma reads as follows: Let $ f \in L^1 (a,b) $ satisfies \begin{equation*} \int^b_a f(t) \varphi'(t) dt =0, \ \ \forall \varphi \in C^{\infty}_0(a,b), \end{equation*} ...
0
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1answer
108 views

Geodesic in half cone [closed]

Consider a right semicircular cone with height $h$ and radius $r$ given by $\mathcal{C}=\left\{\left(\frac{h-u}{h}r\cos(\theta),\frac{h-u}{h}r\sin(\theta),u\right)\,:\,u\in[0,h],\,\theta\in[0,\pi]\...
6
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0answers
154 views

The distributional gradient of the closest isometry to the differential of a smooth map

The setting-a "linear algebra" fact: Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
0
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0answers
56 views

Functional Derivative on Vector Field (Definition)

Given a set $\mathcal{M}$ of functions $\mathcal{X} \to \mathbb{R}$, a functional $J: \mathcal{M} \to \mathbb{R}$ and $f, \phi \in \mathcal{M}$, the functional derivative of $J$ at $f$, $D_f(J)$ is ...
2
votes
1answer
76 views

About the continuity of a function on BV

For a fixed $u \in BV(\mathbb{R}^N)$, consider the function $h:(0,+\infty) \to BV(\mathbb{R}^N)$, given by $h(t) = u (tx)$. Is $h$ continuous?
3
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0answers
146 views

Reference Request: A Set-Valued Minimax Theorem?

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are subsets of $L^2(X,\Sigma,\mu)\cap L^{\infty}(X,\Sigma,\mu)$, where $\mu$ is a finite-measure on $(X,\Sigma)$. Let $F:L^2(X,\Sigma,\mu)\times L^2(X,\...
3
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0answers
79 views

What to do when Euler Lagrange Equation is highly nonlinear ode?

In $\mathbb{R}^3$, suppose there is a curve on X-Y plane $y(x)$ defined on $x\in [-a,a]$ satisfying: $y(x)\geqslant 0$; $y(-a)=y(a)=0.$ Rotate $y(x)$ along x-axis in $\mathbb{R}^3$ and get a solid ...
3
votes
0answers
111 views

Lower semicontinuous and convex envelope

L.Ambrosio, in paper [1] writes: Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...) for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse ...
2
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0answers
71 views

How can I prove that the following function is increasing according to x1?

Suppose that $0 \le {X_1} < {X_2} < {X_3}$ . How is it possible to prove the following function is increasing based on ${X_1}$ in the range of $0 \le {X_1} < {X_2}$ ? $f({X_1},{X_2},{X_3})...
1
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0answers
53 views

A particular semi-linear equation on Riemannian manifolds

Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
4
votes
1answer
255 views

On convex functions which are non constant on every segment

I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
3
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0answers
102 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} - \...
3
votes
1answer
300 views

Solution singular PDE

I've been studying the following singular PDE $$ \mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$ in $\Omega \subset \mathbb{R}^{2}$. Do you know any reference, ...
1
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0answers
107 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) ...
2
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0answers
138 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 ...
0
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0answers
164 views

Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
4
votes
3answers
233 views

Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional $$ f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\}, $$ which ...
3
votes
2answers
138 views

Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
2
votes
1answer
101 views

Uniqueness of minimizers in the Calculus of Variations

Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by $$ f(x,y):= (x^+)^2 + (y^+)^2 $$ where $a^+ = \max\{a,0\}$ for any real number $a$. Given a Lipschitz regular domain $\Omega \...
6
votes
1answer
186 views

Anisotropic perimeter and regularity of anisotropic minimal surfaces

1. Introduction. By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set. Question. What are the known regularity results for ...
0
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0answers
52 views

Elliptic Dirichlet problems with measure boundary data

Can you point out any references on the Dirichlet problem for divergence-type elliptic operators with a Radon measure as boundary data?
1
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1answer
108 views

convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture). Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive ...