Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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2
votes
2answers
223 views

Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\ X(0,x) = x, &x \in \mathbb R \end{cases} $$ where $\chi$ denotes the ...
3
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1answer
124 views

Positive part of Cauchy sequence of sobolev functions is again Cauchy

Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
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0answers
44 views

Hopf type lemma for fractional Laplacian

Let $0<s<1$ and $u\in C^s(\mathbb R^N).$ Does the Hopf type of maximum principle hold for s-super-harmonic function $(-\Delta)^su\geq 0$ in a smooth bounded domain $\Omega\subset \mathbb R^N.$
5
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2answers
2k views

Elementary calculus estimate or not?

Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$ $$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
3
votes
1answer
118 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^...
1
vote
1answer
351 views

Existence for an overdetermined system of PDEs

I am interested in the existence of a vector valued solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are twice more equations than unknowns. More precisely: Let $A$ and $...
2
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0answers
64 views

Trace operators on submanifolds

In the following paper, Sobolev Spaces on Riemannian Manifolds with Bounded Geometry: General Coordinates and Traces https://arxiv.org/abs/1301.2539 The authors prove trace theorems for general ...
7
votes
2answers
275 views

System of linear pde with non constant coefficients

I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even ...
2
votes
0answers
64 views

$W^{2,p}$-estimates for Neumann boundary condition to Poisson equation

Consider the following Poisson-Neumann problem in a lipschitz bounded domain $\Omega\subset \mathbb{R}^3$: $-\Delta u=F,\quad \partial_n u\restriction_{\partial\Omega}=0$. Here $F\in L^p(\Omega)$. ...
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0answers
106 views

Solving partial differential equation by form method (semigroup theory)

We have the equations$\newcommand{\div}{\operatorname{div}}\newcommand{\grad}{\operatorname{grad}}$ $$\rho(x)\frac{\partial^2u}{\partial^2}(t,x)=\div(A(x)\grad u+B(x)\grad \frac{\partial u}{\partial ...
2
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0answers
138 views

Solution to Heat Equation By Projection

Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation $$ \partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x), $$ for some fixed $p\in C^2(\...
1
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0answers
47 views

The ill-posedness of $L^2$-super critical nonlinear Schrödinger equation

For nonlinear Schrodinger equation$$\begin{cases}iu_t+\Delta u+|u|^\alpha u=0\\u(0)=\phi\in H^1(\mathbb R^d)\end{cases}$$ where $\alpha>\frac 4d$. In Christ, Colliander, Tao's paper Ill-posedness ...
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0answers
129 views

Is $(e^{u}+1)\Delta u+u=0$ the Euler-Lagrange equation of a functional energy?

Does there exist a functional energy $I$ such that $$(e^{u}+1)\Delta u+u=0$$ is the Euler-Lagrange equation associated with the energy functional $I$?
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0answers
67 views

Weak Hessian for solutions to certain quasilinear elliptic PDEs

In Chapter 4 of their famous treatise Linear and quasilinear elliptic equations, Ladyzhenskaya and Uraltseva deal with equations of the form $$\sum_{i=1}^n\frac{d}{dx_i}a_i(x,u,\nabla u)+a(x,u,\nabla ...
1
vote
1answer
133 views

Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations

I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801. In Lemma 2.1, the authors said for any $\phi \in L^{\...
2
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1answer
96 views

Reference request: uniqueness for a certain PDE systems

I'm working on a system of the following form: $$(1) \,\,\,\,\,\ \begin{cases} u_{tt} + L_1u + L_2v= f, \\ \nabla u - \nabla v - \nabla v_t=0 \end{cases} $$ where $u(x,t)$ and $v(x,t)$ belong to ...
4
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0answers
128 views

Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$

Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
2
votes
1answer
174 views

Simple existence and uniqueness for second order and linear elliptic PDE

Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$. I am interested on the existence of solution for the following problem: given a continuous ...
2
votes
1answer
147 views

Euclidean type Sobolev inequality on Riemannian manifolds

Let $M$ be a complete, non-compact Riemannian manifold. Let $D$ be a bounded domain with smooth boundary $\partial D$ in $M$. What is the minimum requirement (about domain $D$, curvature,..) so that ...
1
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0answers
72 views

One dimensional periodic travelling waves to some pde

Travelling wave equation on one dimension to Gross Pitaeavkii equation is $$ \phi '' +ic\phi'+\phi (1-|\phi|^2)=0\qquad (1) $$ where $c\in (0,\sqrt{2})$ and $ \phi$ is a complex valued function. I am ...
8
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1answer
171 views

Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?

Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space: $$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
2
votes
1answer
146 views

The Monge- Ampère equation with a non positive right hand side

Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the ...
1
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0answers
40 views

How can we show that a solution depends on more than one variable? [closed]

I have obtained theoretically a solution to a nonlinear Schrödinger-type equation in dimension two. I also proved that is not constant. Now, I wonder if it depends on two variables and not only in one,...
0
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0answers
67 views

Lax-Milgram for Banach spaces

I have the following problem which contributes to my PhD thesis: Conca and Donato showed that in perforated domains $\Omega_\epsilon:=\Omega\setminus\bigcup_{k\in\mathbb{Z}^3} (\epsilon^3 T+\epsilon ...
2
votes
2answers
167 views

Estimate of a solution of Schroedinger equation for a free particle

Let $\psi(x,t)$ be a solution of the Schroedinger on the line $$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$ One assumes that $\psi(x,0)$ "behaves well" as $...
1
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0answers
104 views

Trace and second-order inverse trace on space with Gibbs measure

Consider $(t, x)\in [0,T]\times (\mathbb{R}^d,d\mu)$, where the measure $d\mu(x)=K^{-1}\exp(-U(x))dx$ is a reasonable Gibbs measure (it satisfies a Poincaré or log-Sobolev inequality. One can, for ...
1
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0answers
111 views

Extending solutions to the Dirichlet problem

I got stuck on the following problem. Let $\Delta = \left\{ |z| \leq 1 \right\} \subset \mathbb{C}$ be the unit disk, and let $r$ be a holomorphic function on $\Delta$ which is smooth on $\bar{\Delta}...
2
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0answers
89 views

Justification for uniqueness of solutions to dispersive PDE

For the sake of concreteness, we consider the linear Schrödinger equation $$ \partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x). $$ The solution is typically obtained by taking the Fourier transform ...
1
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0answers
50 views

Extension of a compactly supported pseudo-differential operator

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends ...
3
votes
0answers
90 views

Generalizing the heat kernel approach

I notice a way of solving equations that goes roughly like this: Want to solve $Tu=v$, i.e. hoping to find "$T^{-1}(v)$". $T^{-1}$ might not exist "on the nose", so instead we consider $\int_0^\infty ...
0
votes
0answers
69 views

Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?

$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that. For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
1
vote
0answers
49 views

Energy inequality - wave equation

In J. L. - Lions book Quelques méthodes de résolution des problèmes aux limites non linéaires, the author proves the following lemma: Lemme 6.1: Let w be a function satisfying $w \in L^\infty(0,...
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0answers
89 views

Is there any characterization of polynomials in terms of asymptotic properties of Taylor coefficients? [closed]

My formal question is Let $f(z):=\sum_{n=0}^{\infty} c_n z^n$ be a formal power series. Is there any characterization of polynomials in terms of the asymptotic properties the sequence $(c_n)$? For ...
4
votes
0answers
76 views

Biharmonic heat flow on compact manifolds

Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus. Are there any estimates on the Green function (bihamornic heat kernel), for ...
2
votes
0answers
46 views

Solution of $\vec{p}-$Laplace equation

Let $\Omega \subset {\mathbb{R}^n}$ is bounded domain with smooth boundary. We consider the bvp $$ - \sum\limits_{I = 1}^n {{\partial _{{x_i}}}\left( {{{\left| {{\partial _{{x_i}}}u} \right|}^{{p_i} ...
4
votes
1answer
283 views

Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients

Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below). Given $T>0$ and $n \in \bf Z$, consider the following ...
0
votes
0answers
44 views

On $L^\infty$ norm of solutions to time dependent differential equations

I am new to the theory of differential equations and weak solutions. I am looking for references regarding the analysis of the $L^\infty$ norm of weak solutions to linear second order time ...
3
votes
0answers
71 views

divergence equation with prescribed normal trace

Let $\Omega \subset \mathbb{R}^n$ be a smooth domain and $\nu$ be the outer unit normal to $\partial \Omega$. Given $\phi \in L^{\infty}(\partial \Omega)$ such that $\int_{\partial \Omega} \phi d\...
0
votes
1answer
76 views

Fractional super-harmonic functions

Is this statement true. A bounded half-superharmonic function in $\mathbb R$ is a constant. That is $(-\Delta)^{1/2} u\geq 0$ implies $u\equiv 0.$
1
vote
1answer
71 views

Conservated quantity and hyperbolic equation

Given the hyperbolic Vlasov equation $$ \frac{\partial f }{\partial t} +v\nabla_x f + F(t,x)\nabla_vf =0$$ where $f=f(t,x,v)$ and $(t,x,v)\in \mathbb{ R}\times\mathbb{R}^{n}\times \mathbb{R}^{n} $. ...
0
votes
0answers
42 views

what classical PDEs have analytical expressions for soliton-like shape solutions but motionless?

what classical PDEs have analytical expressions for soliton-like shape solutions but motionless? for example, KdV has analytical expressions of the kind (sech^2(x-vt)), but all of them are ...
1
vote
1answer
74 views

Smooth approximation of a subharmonic function in the barrier sense

Let $f$ be a continuous function on $\mathbb R^n$ such that $\Delta f \ge 0$ at a point $p$ in the barrier sense. More precisely, for any $\epsilon>0$, there exists a smooth function $f_{\epsilon}$ ...
1
vote
1answer
302 views

Laplace spectrum of the $2$-Sphere [closed]

The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this ...
2
votes
0answers
66 views

About p-laplacian and variations

Let $\Omega \subset \mathbb{R^{n}}$ be a domain (open and connected set), for $p\geq 2$, the $p$-laplacian is defined by: $\Delta_p u= \operatorname{div} (|\nabla u|^{p-2} \nabla u)$, in non-...
1
vote
1answer
262 views

Sobolev embedding in the space of continuous functions [duplicate]

Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be ...
1
vote
0answers
47 views

discrete Fourier transform for composition of differential operators on a grid

This question pertains to stability analysis of finite difference methods using the discrete Fourier transform. Suppose I have a convection diffusion equation of the form: (1) $\hspace{.5in}u_t + \...
7
votes
1answer
267 views

Yau's conjecture on nodal sets for manifolds with boundary

I've just read a review paper about Yau's conjecture on nodal sets of the eigenfunctions for the Laplace operator on manifolds. Briefly, if $\phi_\lambda$, $\lambda$ are an eigenpair for the Laplace-...
1
vote
0answers
64 views

Obstacle problems for minimal hypersurfaces

Given a compact Riemannian $n+1$-manifold $M$ with (possibly not mean convex) boundary (smooth or probably with codim $>2$ singularities). Consider the following problems, 1) fix a homology class $...
1
vote
1answer
127 views

Lemma from Donnelly-Fefferman's paper

I am reading the paper Nodal sets of eigenfunctions of the Laplacian on Surfaces by Donnelly and Fefferman available here. I have a problem understanding Lemma 5.10. To my understanding, what follows ...
0
votes
0answers
106 views

The uniqueness of a solution of a differential equation on a unit circle

Trying to solve one problem in the geometry of 2-dimensional Banach spaces, I arrive to the problem of uniqueness of the following differential equation on a function $r:\mathbb T\to(0,1]$ defined on ...

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