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

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9
votes
3answers
532 views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? ...
3
votes
0answers
141 views

ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$. Actually for any solution of the ...
3
votes
0answers
59 views

Semi-continuity of the dimension of the null space

Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...
0
votes
1answer
159 views

A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction, $$\Delta u=-\lambda u.$$ I was wondering under what condition (for example, spaces ...
7
votes
2answers
249 views

Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
0
votes
0answers
135 views

Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
2
votes
2answers
133 views

Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me. Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...
2
votes
1answer
209 views

Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and $$ \mathrm{Ricc}(g)=\lambda g, $$ $h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
2
votes
1answer
113 views

A question about viscosity solutions

Let $A:=[a,b]$ be a closed interval in $\mathbb{R}$. Let $F(x,p,q,r)$ be a function from $[a,b]\times \mathbb{R} \times \mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}$ describing a second order ...
2
votes
0answers
55 views

Hypoelliptic pseudodifferential operators and Fredholm equations?

I read here the following: The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable ...
2
votes
1answer
40 views

Literature on Lateral cauchy problem

Can you please provide me with books that deal with lateral cauchy problem? Also introductory articles are fine by me. Is this topic covered in books like Evans', Taylor's, Hilbert-Courant's or ...
1
vote
1answer
114 views

Are solutions of the Beltrami Equations necessarily smooth?

Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The Beltrami Equations are defined as the following system of PDE’s on the domain $ ...
2
votes
0answers
37 views

elliptic regularity of Neumann problem on Square

I asked a similar question the other day, but I will be more precise now. Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider $$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) ...
2
votes
0answers
71 views

elliptic regularity for Neumann BVP on square

I am interested in the regularity of ellitpic equations like $$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ ...
4
votes
0answers
60 views

Minimisers and critical points of variational integrals

In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral: \begin{equation*} ...
16
votes
4answers
995 views

Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary. Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known? An important example is the $n$-sphere ...
0
votes
0answers
34 views

regularity for an elasticity problem

Consider $B$ a bounded domain of $\mathbb{R}^3$ with a smooth boundary $\partial B$. Is there any reference for the optimal regularity with respect of the boundary data of the solution of the ...
5
votes
3answers
268 views

A space of distributions vanishing on the boundary

The revised question After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let $$\mathcal ...
4
votes
1answer
126 views

Mixed norm estimate for the heat equation

Consider the inhomogeneous linear heat equation $$\partial_tu-\Delta u=F$$ on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice ...
4
votes
1answer
181 views

Zeta-Determinant Theorem

Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here. When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...
4
votes
1answer
129 views

General solution to null-divergence equation

I am interested in whether each component of a divergenceless vector can itself be written as a divergence. My motivation for this question is the characterization of so-called trivial conservation ...
6
votes
3answers
227 views

Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators. So my question is something like this: Let $i \partial_t \psi(x,t) = ...
2
votes
0answers
101 views

Changing frames of the tangent bundle with Schwartz functions [closed]

Let's consider two global frames $\{v_{1},....v_{N}\}$ and $\{u_{1},....u_{N}\}$ of the tangent bundle $T\mathbb{R}^N$. Now consider the matrix $\{f_{i,j}\}$that change the frame $\{v_k\}$ to ...
2
votes
1answer
135 views

Connection between the p and q Laplacians

I'm just looking for some quick and dirty intuition(and/or reading material) about the following: I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( ...
0
votes
0answers
59 views

Modified Sine-Gordon on Torus

Has the modified (sign altered) Sine Gordon Equation: $$ \alpha_{ss} + \sin \alpha = 0 $$ been found to have interesting geometrical properties? With symmetric ODE I obtained closed loops on a ...
2
votes
2answers
206 views

Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian ...
0
votes
1answer
117 views

Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of: $$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$ $$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ...
5
votes
1answer
175 views

Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...
2
votes
0answers
53 views

Using compactness method to prove the existence of a pathwise solution to an SPDE

For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
2
votes
1answer
154 views

Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
6
votes
0answers
146 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
4
votes
0answers
105 views

Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
1
vote
0answers
72 views

Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} ‎\Delta^2u=‎‎\lambda ‎‎\dfrac{u}{|x|^4}‎‎+u^{‎p}‎ & ...
0
votes
0answers
38 views

Continuous time dynamic programming: Quadratic guess for value function

In a control problem like so: $$J = min \int_0^{t_f} Qx^2 + Ru^2 dt $$ $$\dot{x} = Ax + Bu$$ $$x(0) = x_0$$ The regular Linear Quadratic Regulator is attained by asssuming that the optimal value ...
3
votes
1answer
191 views

Theorems that tell if an explicit analytical solution is possible for nonlinear PDEs

Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods? Where analytical methods I refer to methods such as power series or any methods that use ...
0
votes
1answer
69 views

Reference request for the focussing example

I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. ...
3
votes
0answers
67 views

Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional $$ F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1 $$ where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
1
vote
0answers
80 views

Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$. Let ...
3
votes
1answer
154 views

Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants. Let $Q(R) := Q(R^2,R) = B(0, R) \times ...
4
votes
2answers
297 views

Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks
2
votes
1answer
160 views

Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where. Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm $$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...
2
votes
1answer
113 views

Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation $$ u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1} $$ for a Hölder continuous coefficient $a(x,t)$ satisfying $$ 0<C_o \le ...
3
votes
1answer
53 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet. Link:http://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded ...
2
votes
0answers
58 views

Reference/proof for parabolic Holder spaces property

Suppose that a function $u:[0,1]\times[0,T]\to\mathbb{R}$ belongs to the parabolic Holder space $C^{2+\alpha,1+\alpha/2}$, for $\alpha\in(0,1)$. What can be said about $u_x=\partial_x u$? I am not ...
5
votes
0answers
222 views

The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$. We consider the following polynomial vector field on ...
0
votes
0answers
35 views

The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that ...
1
vote
0answers
54 views

Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...
1
vote
0answers
55 views

Level Set Advection with Extension Velocity

We're studying the following system of PDEs for a scalar function $F(x, t)$ with $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$. The function $F(\cdot, t)$ is a level set function for a time-dependent ...
3
votes
1answer
133 views

Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$

I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space. I am looking for references about results for the scalar case ...
0
votes
1answer
69 views

Does this time-dependent trace space have a name?

This question is a follow up to this question. Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in ...