Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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17
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0answers
409 views

Why is there a connection between enumerative geometry and nonlinear waves?

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it. Recently I encountered in a class the fact that there is a generating function of ...
1
vote
0answers
97 views

Focusing NLS: $L^2$ convergence of a solution as $t\rightarrow +\infty$

Consider the cubic focusing non linear Schrodinger equation in dimension $n\geq 2$: $$(iu_t+\Delta)=-|u|^2u\qquad u(0,x)=u_0(x)\in L^2(\mathbb{R}^n)$$ Can we find an initial data $u_0\neq 0$ such ...
1
vote
1answer
99 views

Image of the trace operator on W^{1,1}

Let $\Omega \subset R^n$ be a bounded region with Lipschitz boundary. Is the trace operator $T: W^{1,1}(\Omega)\rightarrow L^1(\partial \Omega)$ surjective? If not, what is the image?
1
vote
3answers
241 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
2
votes
2answers
154 views

Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
0
votes
0answers
148 views

Can a function be constructed from the direction of its gradient?

Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation $Du=J|Du|$, $u|_{\partial \Omega}=f$ has a solution in a ...
1
vote
0answers
99 views

localization of the $L^p$ variation for heat equation

I'm struggling with yet another question for the classical heat equation in the whole space $R^d$. This question seems basic at first sight, but I think it is nontrivial in the end so here it is. The ...
0
votes
1answer
164 views

Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness

Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$. Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...
1
vote
1answer
199 views

Sobolev Inequality

Let $\Omega$ be a bounded region in $R^n$ and define $W:=\{ u \in H^{1}(\Omega): u(x_0)=0 \},$ where $x_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that ...
1
vote
0answers
124 views

Extension of solutions of PDE

Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed. Let ...
10
votes
1answer
286 views

Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs. Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...
1
vote
2answers
237 views

How to construct a harmonic function with non-zero gradient on manifold with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole $M$. More ...
2
votes
0answers
124 views

How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...
0
votes
1answer
106 views

weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions

I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, ...
3
votes
0answers
193 views

is this a known method for solving PDEs

I recently posed a system of PDEs to solve on MSE at http://math.stackexchange.com/q/514147/36530. It was quickly solved by a nice pair of subsitutions. However, in this post, I'd like to show here ...
1
vote
1answer
84 views

Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$

Let $D\subset R^d$ be a bounded Lipschitz domain. We know that the Neumann eigenfunction lies in $C(\overline{D})$ (i.e. continuous up to the boundary). This can be seen from the fact that ...
1
vote
1answer
145 views

functions of bounded variation and gradient vector measure

I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that $$ \lim_{r\rightarrow 0}\frac{C}{r^{n+1}} ...
2
votes
1answer
117 views

A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1 $$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) ...
7
votes
3answers
191 views

Spectrum of Dirichlet Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and ...
0
votes
1answer
155 views

carleman inequality

Is there a connection between carleman inequality discovred by T. Carleman in 1922 if I am not mistaken in his research on quasianalytic functions and what is called Carleman estimates used in the PDE ...
1
vote
2answers
154 views

vector valued pde's good reference

I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows. Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n ...
1
vote
0answers
38 views

parabolic PDE with pseudomonotone operators

I am looking for a reference where well-posedness of problems $$u_t + A(t)u = f$$ is addressed via the Galerkin method where $A$ is a pseudomonotone operator. I am aware that Roubicek's book ...
4
votes
1answer
262 views

Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support. Is the reverse true? Namely that if some PDO ...
14
votes
4answers
648 views

When to use more exciting function spaces than ordinary Sobolev spaces?

In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions. For example, Sobolev spaces ...
0
votes
0answers
129 views

If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?

Let $\phi_n$ be a sequence of mollifier converging to the identity $$ \phi_n(x) \to \delta_{0}(x), \text{pointwise}, $$ with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in ...
1
vote
1answer
170 views

Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$ Further assume a solvability condition $$\int_\Omega f ...
0
votes
0answers
64 views

PDE with periodic solution with Galerkin method

Can anyone give me a reference to where I cacn find existence of solutions to parabolic PDE (eg. a standard linear PDE) with initial and end conditions $u(0) = u(T) = u_0$ so that periodic solutions ...
2
votes
0answers
121 views

W^2,p regularity for solutions of elliptic equations

I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...
2
votes
1answer
238 views

Fully non-linear PDE

A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...
1
vote
1answer
99 views

radial limits of subharmonic functions

Let $u$ be a non-negative subharmonic function on the unit ball in $\Bbb{R}^n$. Does it follow that there exists a radial limit (including limits of infinity or negative infinity) along almost every ...
2
votes
0answers
91 views

Why pseudoconvexity is important in Partial differential equation theory?

I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
2
votes
0answers
145 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
1
vote
1answer
149 views

Blow up of solutions to parabolic PDEs

I am looking for a text or answer detailing the blowup of solutions to parabolic PDE (eg. heat equation) in Sobolev space setting. I heard blowup is related to size of domain but I can't find any nice ...
1
vote
0answers
137 views

Solving PDE problems by introducing special function spaces [closed]

When dealing with Partial differential equations, appropriate choice of the space may sometimes be very useful. The most frequently used spaces include $L^p$, $H^p$, $W^{k,p}$ and Besov spaces. For ...
2
votes
2answers
125 views

Changing the space of test functions in PDEs

Let $V \subset H \subset V'$ be a Hilbert triple. We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying $$\int_0^T u(t)\varphi'(t)=-\int_0^T ...
1
vote
1answer
160 views

What are the advantage of using operational calculus for numerically solving pde compared to FE or FD?

For numerically solving a partial differential equation (PDE) what advantage does operational calculus (OC) has over common methods like finite difference (FD), and finite element (FE)? I mean OC in ...
3
votes
1answer
186 views

Boundary flux maximizing drift (velocity) vector fields for 2D heat equation

Looking for literature / known results on the following class of problems: Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar ...
4
votes
1answer
178 views

Null sets in PDE

Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = ...
4
votes
1answer
133 views

diffusions corresponding to estimators

I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
13
votes
5answers
901 views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
3
votes
1answer
209 views

iwaniec's conjecture

Does anyone know whether there is any geometric applications of the iwaniec's conjecture on $ l^p $ bound of beurling alfhors transform( or the complex hilbert transform). One application could have ...
4
votes
3answers
404 views

Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question: "Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?" I believe the answer to ...
1
vote
1answer
158 views

Does a particular iteration produce a weak solution to a non linear pde?

Consider the following non linear pde in the unknown $v(x,y)$: $$ \frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$ where $t$ is some fixed small ...
2
votes
1answer
256 views

Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...
6
votes
1answer
116 views

Ergodic Mean for Schrodinger flow

Let us consider the linear Schrödinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ be its ...
4
votes
0answers
130 views

Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function $\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted $L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions $f: ...
1
vote
1answer
129 views

Nonlinear parabolic PDEs existence with Galerkin method?

Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...
10
votes
2answers
354 views

the spectrum of the Laplacian and Dirac operator on $S^3$

A paper on supersymmetry in 3-dimensions uses results on the spectra of elliptic operators on $S^3$: The eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ ...
4
votes
1answer
165 views

Local boundedness of weak solutions of heat equations…?

(Please, what is going on?) The following claim is from a paper which was apparently reviewed by László Erdös, Zhongwei Shen, and Bernard Heffler. Someone tell me it's true. Surely it's true. The ...
1
vote
2answers
95 views

Bound deg 3 partial differential operator on Laplace eigenfunction?

I am no expert on PDE and analysis but I am looking for certain technique from PDE. Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...