**2**

votes

**0**answers

323 views

### Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like
$$
\partial^2\phi+V(\phi)=\delta^D(x).
$$
I do not know if a real ...

**2**

votes

**0**answers

107 views

### Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...

**2**

votes

**0**answers

102 views

### Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...

**2**

votes

**0**answers

163 views

### Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation
$$u_t-iu u_x-iu_{H,xx}=0$$
where $u_H(x,t)$ denotes the ...

**2**

votes

**0**answers

83 views

### P-laplacian equation

Hi guys,
in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ?
Thank you!

**2**

votes

**0**answers

139 views

### Extension divergence-free, curl-converging vector field

Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...

**2**

votes

**0**answers

196 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**2**

votes

**0**answers

59 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II

This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
...

**2**

votes

**0**answers

101 views

### Best constant of Gagliardo-Nirenberg inequality in exterier domain

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have
$||u||_4^4\leq C||u||_2||\nabla ...

**2**

votes

**0**answers

268 views

### How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements?

I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This ...

**2**

votes

**0**answers

208 views

### Recovering full regularity by energy method in the heat equation

Consider the heat equation
$$
u_t = u_{xx} + f,
$$
on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...

**2**

votes

**0**answers

144 views

### A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...

**2**

votes

**0**answers

81 views

### Asymptotics of quasilinear elliptic equations with Dirac right hand side

On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form
$$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha ...

**2**

votes

**0**answers

155 views

### Integrability of ground state solution for elliptic equation

For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By ...

**2**

votes

**0**answers

265 views

### Variational Formulation of Boundary Value Problems With Unknown on the boundary

Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$,
\begin{eqnarray}
Lu &=& \frac{\partial u}{\partial t},
u(x,0) &=& u_0 \; \; ...

**2**

votes

**0**answers

181 views

### Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that ...

**2**

votes

**0**answers

220 views

### Equation of motion of a charged string in a twisted torus.

The flat torus background:
Say we want to study the sigma model of strings (closed strings $S^1= \mathbb{R}/\mathbb{Z}$) on a flat torus (for example $T^3=\mathbb{R}^3/\mathbb{Z}^3$ with a flat ...

**2**

votes

**0**answers

449 views

### Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background
Consider the one dimensional second order elliptic PDE,
$$
\left\{\!\!
\begin{aligned}
& -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\
& u(0)=u(1)=0
\end{aligned}
...

**2**

votes

**0**answers

146 views

### Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?

This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = ...

**2**

votes

**0**answers

264 views

### A free boundary problem by finite difference method

I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...

**2**

votes

**0**answers

260 views

### General solutions for HJB equations in a special case.

I am reading the book of Wendell Flemming in control theorem to learn the HJB equation
Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that ...

**2**

votes

**0**answers

286 views

### Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...

**2**

votes

**0**answers

273 views

### Poisson problem with a “scaled” Laplacian.

Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form
$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...

**2**

votes

**0**answers

217 views

### Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...

**2**

votes

**0**answers

212 views

### Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?

Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary ...

**1**

vote

**0**answers

81 views

### A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...

**1**

vote

**0**answers

47 views

### presence of turbulent phenomena in systems of linear pde?

Are there linear systems of PDE that are known to have solutions which exhibit turbulence, or can turbulence be firmly classified as a fundamentally non-linear phenomenon, similar to solitons or shock ...

**1**

vote

**0**answers

32 views

### Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question:
Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that
$F(x,\cdot)$ is convex with respect to the second variable.
$F(\cdot,v)$ ...

**1**

vote

**0**answers

80 views

### Half Laplacian; (definitions of) and regularity

I have a question regarding the half Laplacian $ (-\Delta )^\frac{1}{2}$ on some smooth bounded domain $ \Omega$ in $R^N$. I am attempting to clarify some confusion with the various definitions. ...

**1**

vote

**0**answers

81 views

### Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< ...

**1**

vote

**0**answers

81 views

### Stokes operator without dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known :
$\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $
$f \rightarrow u$ such that $ - \Delta u = f $
where $ ...

**1**

vote

**0**answers

97 views

### Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact.
Is it true that the embedding $H^1_0(\Omega) \rightarrow ...

**1**

vote

**0**answers

69 views

### $L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...

**1**

vote

**0**answers

70 views

### Comparison principle using truncation for porous medium equation

For a porous medium equation (eg. $u_t - \Delta \Phi(u) = f$), is it possible to obtain a comparison principle for very weak solutions (eg. if $u_0 \geq 0$ and $f \geq 0$ then $u \geq 0$ a.e.) using ...

**1**

vote

**0**answers

148 views

### Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty...
My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...

**1**

vote

**0**answers

44 views

### Well-posedness of a certain linear Cauchy-problem

I am interested in solutions to the linear Cauchy problem
$$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, ...

**1**

vote

**0**answers

78 views

### A question on the maximum principle for Schrodinger operators

Let $(M^n,g)$ be a closed Riemannian manifold, and $L=-\Delta+V$ be a Schrodinger operator, $V\in C^{\infty}(M)$. In answers to the two questions (First eigenvalue of SchrÃ¶dinger operator is simple 1 ...

**1**

vote

**0**answers

103 views

### Estimate the smallest eigenvalue of a Schrodinger operator

There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...

**1**

vote

**0**answers

80 views

### On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; ...

**1**

vote

**0**answers

70 views

### Limit Toward Discontinuous Point of Dirichlet Boundary Value

The question arises from a paper on Schwarz's domain decomposition method (click here).
We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below.
Now we ...

**1**

vote

**0**answers

45 views

### Lyapunov Schmidt; basic example

I am attempting to understand the Lyapunov-Schmidt method with a simple example but I am running into trouble. Here is the example I am considering. Suppose $ v>0$ satisfies $ -\Delta v - v=0 $ ...

**1**

vote

**0**answers

95 views

### Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)

Denote by $\mathbb{E}(g)$ the solution of the PDE
$$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$
$$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$
Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...

**1**

vote

**0**answers

35 views

### Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?

Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?
Where $\omega\subset\subset \Omega$ with $\Omega$ is a $C^2$ ...

**1**

vote

**0**answers

100 views

### on high order Laplacian

Roughly speaking, we have good understanding of the solution to heat equation $u_t-\Delta u=0$, on bounded or unbounded domain. For example, we know the decay rate, we know it generates analytic ...

**1**

vote

**0**answers

85 views

### Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...

**1**

vote

**0**answers

101 views

### Passing to the limit in a PDE (subsequence problems)

For $w \in L^2(0,T;H^1)$, consider the PDE
$$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$
where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...

**1**

vote

**0**answers

113 views

### Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...

**1**

vote

**0**answers

110 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**1**

vote

**0**answers

93 views

### How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?

(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...

**1**

vote

**0**answers

78 views

### Regularity of weak solutions for a quasilinear problem

Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in ...