0
votes
0answers
64 views

Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
3
votes
0answers
39 views

Regularity of a Weak Solution

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
1
vote
0answers
20 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
0
votes
1answer
60 views

Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
0
votes
1answer
60 views

Eikonal Equation [duplicate]

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $A \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=A$ and ...
0
votes
1answer
24 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
0
votes
1answer
34 views

An application of the mountain pass lemma

I am trying to show the existence of classical solution for the following problem using the mountain pass theorem : $$ \left\{ \begin{array}{ccccccc} u^{''} + \lambda u + u³ = 0 (0<t<\pi)\\ ...
1
vote
0answers
35 views

Semilinear Poisson Equation Using Direct Method of Calculus of Variations

The following problem comes from: http://people.physics.anu.edu.au/~gvn105/analyticMethPDE.pdf 12.9 Exercises 12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be ...
2
votes
1answer
40 views

Differential of Lagrangian

My professor wrote this $\frac{\partial L}{\partial q}\dot{q}=\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})$. Due to the fact that I am very very very very bad at Math, could you explain me about ...
0
votes
1answer
38 views

Elliptic partial differential equations

Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
0
votes
1answer
27 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
1
vote
0answers
21 views

specific non linear pde

I would really appreciate to hear your insights or comments about the following problem: Consider the following non linear pde: let $\Omega$ be the unit square with vertices at (0,0),(1,0),(0,1) and ...
0
votes
0answers
26 views

How to find the infintesimal generator and conserved current of the symmetries of the minimal surface problem

For the Lagrangian $L(x,y,z,z_x,z_y)=\sqrt{1+z^2_x+z^2_y}$ how do you find infinitesimal generator and conserved current of the six symmetries (3 translations and 3 rotations)? I was using Noether's ...
1
vote
1answer
22 views

Derivation of inner variations

In Giaquinta's and Hildebrandt's 1996, "Calculus of Variations 1", pages 147-148, they develop the definition of inner variations. They first fix $\lambda\in ...
2
votes
1answer
51 views

Palais-Smale conditions of functional involving noncoercive differential operator

I am working on mountain-pass like theorems for the problems $$ - u_{xx} - a u = \pm |u|u+|u|^2u , \ x \in (a,b), \quad u(a)=u(b) = 0$$ where $a \in L^\infty((a,b))$ is positive (I take the one ...
0
votes
1answer
62 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
1
vote
1answer
52 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
3
votes
1answer
46 views

Euler-Lagrange on restricted set

I am reading a chapter of Evan's book on weak convergence methods for nonlinear PDE's p.49 and it states that the Euler Lagrange equation for the functional \begin{equation} I[w]:=\int_U|Dw|^2 ...
0
votes
1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
6
votes
0answers
89 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
2
votes
2answers
109 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
1
vote
1answer
30 views

Derivation of weak form of Euler Lagrange Equation

In Giaquinta's and Giusti's 1982 paper entitled "On the regularity of the minima of variational integrals", they look at the following quadratic functional: \begin{equation} ...
1
vote
2answers
53 views

Existence of solution in Hölder spaces

Let's say we have a PDE, for example the Laplace equation: $$ \Delta u = f. $$ Usually, to solve such a thing, one finds its variational formulation, and solves it in some Sobolev space. My question ...
1
vote
1answer
61 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
3
votes
1answer
81 views

Regularity for this variational problem

The Problem. Assume $\Omega \subset \mathbb{R}^2$ bounded and $u \in H^1(\Omega,\mathbb{C})$ is some fixed function. Now consider the variational problem $$ F_\lambda(v) = \frac{\lambda}{2} ...
2
votes
2answers
65 views

Showing that this Coercivity condition implies uniform boundedness of a minimising sequence.

The following problem is in Dacorogna's book "Introduction to the Calculus of Variations": Let $\Omega\subset\mathbb{R}^n$ be open and bounded with a Lipschitz boundary. Let $f\in C(\mathbb{R}^n\times ...
2
votes
1answer
220 views

Does weak lower semi-continuity follow from convexity for complex valued functions?

I have the following problem: The definition if weak lower semicontinuity states A function $I[\cdot]$ is weakly lower semicontinuous on $W^{1,q}(U)$ provided \begin{equation} I[u] \leq ...
6
votes
2answers
1k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
8
votes
2answers
356 views

Finding Euler-Lagrange equations

Maybe you can help here. There is kind of a lengthy setup to understand what the question is asking. There is a paper I'm reading, and in one section of it I can't make heads or tails of the result. ...
1
vote
1answer
95 views

Why is the weak limit of the derivatives the derivative of the weak limit here?

In [1, chapter 8.2.1.b, p.466] the author uses the following argument: Let $U \subset \mathbb{R}^N$ be an open, bounded domain with smooth boundary. Given a bounded sequence $(u_k)_{k \in ...
4
votes
1answer
294 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
4
votes
1answer
228 views

Step in derivation of Euler-Lagrange equations of motion

From http://www.mathpages.com/home/kmath523/kmath523.htm Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we ...
1
vote
0answers
43 views

How to show that this solution is a minimum.

I am reading this paper from Peter Hess: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differential Equations 6 (1981), no. 8, 951–961. In the last page ...
1
vote
0answers
105 views

Null Lagrangians and “Local Degree”

Let $u: U\subset\mathbb R^n \rightarrow \mathbb R^n$ be a smooth function, $U$ bounded. Let $x_0$ and $r$ be such that $B_r(x_0)$ is disjoint from $\partial U$. Let $\eta$ be a smooth bump function ...
4
votes
1answer
190 views

Do continuous functions preserve weak*-convergence?

I am trying to comprehend the proof of a theorem in the calculus of variations (sketch: the functional $\int\limits_\Omega f(Du_j(x))~dx$ is weak*-sequentially semicontinuous if and only if $f$ is ...
2
votes
1answer
67 views

Pf. of weak lower semicontinuity for convex Lagrangians

This question is about the proof of Theorem 1 in Chapter 8 of Evans' PDE book (p. 468 in the 2nd edition). Let $u,u_k\in\mathrm{W}^{1,q}(U)$ for all $k\in\mathbb{N}$, $U\subset\mathbb{R}^n$ be open, ...
2
votes
1answer
194 views

Finding the Euler-Lagrange operator $\mathcal M$ of a functional $\mathcal F$

I'd appreciate some help with the following problem: Let $F = F(x, \{p_\alpha\}_{|\alpha|\le m})$ be a smooth function of the variables $x\in \overline \Omega$, and $p_\alpha \in \mathbb R, ...
2
votes
0answers
84 views

First Weighted Eigenvalue of the Laplacian

Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$: $$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...
4
votes
0answers
113 views

Nonlinear BVP pde and variational inequality

Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t $$-\Delta u_a + \frac{m(u_a)}{a} = f$$ where $m(r) = \begin{cases} r ...
2
votes
1answer
234 views

How do I obtain an appropriate energy functional from the weak formulation of a partial differential equation?

I'm reading a textbook example on the finite element method: $\nabla^T[D(x,y,z)\nabla u] - a(x,y,z)u + f = 0 $ in R $\partial R= \partial R_1 \bigcup \partial R_2$, $\partial R_1 \bigcap ...
1
vote
1answer
220 views

Monge Ampere and Calculus

I am learning about mass transportation theory and the Monge-Ampere equation, to transport a function $f$ toward $g$ by a change of variable $T$. In particular, in order to solve for : $$ \min \int ...
2
votes
0answers
127 views

Positive rotational symmetric solution for p-Laplacian

I have the the following problem and I just can't get my head around how to solve it. Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
1
vote
0answers
264 views

Book Recommendation Needed: Gradient Descent, Euler-Lagrange

On a lecture note I read about Calculus of Variations faculty.uml.edu/cbyrne/cov.pdf the author talks about Euler-Lagrange equation, then continues to say "unfortunately, many times a closed form ...
5
votes
1answer
700 views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9... There ...
5
votes
2answers
367 views

minimizing the norm of a curl over a domain

According to my computations: The function which minimizes $$\int_\Omega \|\operatorname{curl} f\|^2\,dx$$ should satisfy $$\operatorname{curl}(\operatorname{curl}f) = 0$$ everywhere on $\Omega$, ...
7
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...