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

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4
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1answer
175 views

Progress on isospectral plane domains

Has there been any progress on the smooth isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit ...
4
votes
1answer
193 views

Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...
5
votes
1answer
74 views

Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$. Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...
4
votes
0answers
349 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
3
votes
2answers
659 views

Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...
1
vote
0answers
26 views

Regularity of a flux induced by a potential

Take $\Omega\subset R^n$ with smooth boundary (take a ball for example) a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$ a ...
1
vote
0answers
74 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
0
votes
1answer
88 views

The monotone operator in $BV$ space

I am considering the following minimizing problem: $$ \min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\} $$ where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
2
votes
1answer
84 views

Getting an estimate of the form $\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$ on solution of PDE

Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for ...
2
votes
0answers
38 views

Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
0
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0answers
46 views

The properties of the solution pf minimizing problem with different parameters

I asked an similar problem before but received no respond. Here I modified the problem, add in more informations and assumptions, and with an extra question... Let $\Omega\subset \mathbb R^2$ be open ...
2
votes
0answers
62 views

PDE Parameter-Dependent Center Manifolds

In the book Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems by Mariana Haragus, parameter-dependent center manifolds are discussed. Here it is assumed ...
9
votes
5answers
506 views

List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
0
votes
1answer
76 views

Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says: Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$. ...
3
votes
1answer
202 views

stability of the Monge-Ampère equation

Is there any hope to prove this conjecture (or a similar one)? Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} ...
2
votes
1answer
102 views

Elliptic regularity with mixed boundary conditions

I'm looking for some results about elliptic regularity with mixed boundary conditions. I know they exist with non mixed boundary conditions but where can I find some results for the mixed case? Thanks ...
3
votes
0answers
99 views

Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations: $(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$ $(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$ for ...
0
votes
0answers
171 views

Boundary conditions in the Finite Element Method

I just want to solve a Sturm-Liouville problem in 1D, i.e., \begin{align} (p(x)u'(x))'+q(x)u(x) = f(x) \end{align} with boundary conditions \begin{align} u(0)=a \hspace{1cm} u'(0)=b \end{align} How do ...
0
votes
0answers
61 views

How Minimal solution is obtained as limit of approximations

I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this: I can understand the proof, but I don't know why this solution which is obtained as a ...
0
votes
1answer
69 views

Existence and estimates of a solution of a perturbed first order partial differential equation

My question is as follows: Let $A=\partial_x-\frac y2\partial_z$, $B=\partial_y+\frac x2\partial_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you ...
2
votes
0answers
34 views

Does the fast diffusion equation (or singular PME) on a manifold lose mass if the exponent is small enough?

Consider the singular porous medium equation $$u_t - \Delta (|u|^{m-1}u) = 0$$ $$u(0)=u_0$$ given $u_0$ bounded, where $m \in (0,1)$. When posed on $\mathbb{R}^n$, it is well known that mass is ...
4
votes
1answer
90 views

Elliptic regularity for two dimensional domains

Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to $$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$. If $ f \in ...
5
votes
0answers
98 views

Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory? I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...
4
votes
1answer
93 views

Regularity up to the boundary for the Poisson problem

It seems that the following assertion is widely accepted: For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...
2
votes
1answer
166 views

comparing Laplacian and gradient of function on boundary

Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$. Suppose one knows that there is some $C_1,C_2 \in R$ such that $ x ...
5
votes
1answer
238 views

harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...
1
vote
1answer
72 views

degree theory for elliptic equations; special solutions

I am interested in using degree theory to examine some semilinear problems. But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets ...
0
votes
0answers
71 views

A heat equation approach to the perturbation of vector field with center

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it. We consider the heat equation $$U_{t}=\Delta ...
8
votes
1answer
517 views

Asking for Advices for Choosing a Ph.D thesis problem (in PDE area)

I'm a first year phd student in Germany. I've started my phd study one year ago and I'm currently confused about the topic I've chosen. The program is in the area of PDEs, and actually I didn't learn ...
2
votes
0answers
67 views

Regularity of $u$ in $u_t - \Delta \beta(t,u) = f$, can we get $u_t$ is a function?

I'm looking for reference discussing the regularity of the weak solution $u$ to the equation $$u_t - \Delta \beta(t, u) = f$$ $$u(0) = u_0$$ where $\beta(t,\cdot)$ is a nonlinear function depending ...
0
votes
0answers
34 views

Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution to ...
2
votes
1answer
184 views

Proving compatibility of two Partial differential equations

Given two PDE(s): $F(x,y,z,p,q)=0$ and $G(x,y,z,p,q)=0$ In I.A.N Sneddon's "Elements of Partial Differential Equations",If every solution of $F=0$ is a solution of ...
3
votes
0answers
53 views

Steklov averages in PDE: what to do when we have time-dependent elliptic operator

One may have an equation (with boundary conditions omitted below) $$u_t - Au = f$$ $$u(0)=u_0$$ which has a weak solution $u \in L^2(0,T;V) \cap C([0,T];H)$ in the sense that $$-\int_0^T \int_\Omega ...
9
votes
1answer
216 views

Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field. Does there exist a conformal factor $c$ ...
1
vote
0answers
96 views

A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
1
vote
1answer
248 views

The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

I am interested in solving the following biharmonic eigenvalue problem. $$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\ ...
9
votes
3answers
554 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
147 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
60 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
163 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
277 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
138 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
140 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
221 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
126 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
63 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
116 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
42 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
76 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 $ ...