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

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12
votes
0answers
393 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
10
votes
0answers
165 views

Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates ...
9
votes
0answers
385 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
7
votes
0answers
246 views

Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...
7
votes
0answers
148 views

Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that $||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...
7
votes
0answers
170 views

Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$. C.D. Sogge proved that we have the ...
7
votes
0answers
348 views

Regularity of solutions to a linear degenerate parabolic pde

I've encountered the following problem which is causing me some trouble : Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function ...
7
votes
0answers
260 views

Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
6
votes
0answers
138 views

Local solvability of nonlinear elliptic boundary value problems

Malgrange proves the following statement regarding local solvability of (determined or underdetermined) nonlinear elliptic systems: Let $F_i(x,D^\alpha u)=0$ be a nonlinear elliptic system of order ...
6
votes
0answers
339 views

Lax Pairs for Linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
6
votes
0answers
284 views

Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by $$\frac{\partial}{\partial t} ...
6
votes
0answers
284 views

Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature. For each $s>0$, I have a ...
6
votes
0answers
378 views

How to estimate isoperimetric constant?

Suppose $(X^m, g)$ is a closed Riemannian manifold of dimension $m$ with the following properties, There is a constant $\kappa$ such that $\kappa r^m \leq Vol(B(x, r)) \leq \kappa^{-1} r^m$ for ...
6
votes
0answers
246 views

Dirichlet-to-Neumann map on $C^{k,1}$ domains

I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in ...
6
votes
0answers
808 views

Harmonic maps into compact Lie groups

Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck, Schoen, Richard; Uhlenbeck, ...
6
votes
0answers
365 views

A non-elliptic PDE

I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X : $\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial ...
5
votes
0answers
104 views

Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential $$ I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy. $$ The classical Hardy-Littlewood-Sobolev ...
5
votes
0answers
68 views

Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)? To be more detailed: if I want to show that some ...
5
votes
0answers
89 views

behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses. My interest is in how $ ...
5
votes
0answers
131 views

Pseudodifferential operators on compact manifolds with boundary

I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
5
votes
0answers
168 views

Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
5
votes
0answers
242 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
5
votes
0answers
92 views

$L^p$ regularity for wave equations with coercive boundary conditions

Suppose we have the wave type equation $$\partial^2_tu - L u = 0$$ on a compact manifold with boundary, where $L$ is a second order strongly elliptic operator with coercive boundary conditions (not ...
5
votes
0answers
336 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
5
votes
0answers
280 views

Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
5
votes
0answers
164 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: ...
5
votes
0answers
95 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
5
votes
0answers
344 views

“Euler system” in Christodoulou's The Action Principle and PDEs

In The Action Principle and PDEs Christodoulou spends some time describing what he calls the Euler system associated to a system of variational PDEs (sections 2.5-7, 6.2). Briefly, given a bundle ...
5
votes
0answers
452 views

Any similar inequality in literature?

I got the following inequality: $B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...
4
votes
0answers
105 views

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...
4
votes
0answers
139 views

Nash's proof of De Giorgi-Nash-Moser theorem

I saw this question, but I think the answer didn't fully address what I want to know about it: Nash's paper on parabolic equations. It says almost everything developed later in elliptic and ...
4
votes
0answers
60 views

How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$). Let $f\in ...
4
votes
0answers
66 views

A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting: we have a non-negative supersolution $h$ of the following elliptic equation: \begin{equation} \Delta h + \|\nabla h\|^2 + f(x) \geq 0 ...
4
votes
0answers
280 views

A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...
4
votes
0answers
93 views

Continuity of the curve-shortening flow with respect to the curve

The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in ...
4
votes
0answers
38 views

On a continuous extension of a linear 2nd order PDE

Consider an elliptic (hyperbolic) equation $A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = 0$ in a bounded open plane set $D$, with real-valued functions $A$, $B$, and $C$. Is it true that at least ...
4
votes
0answers
93 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
4
votes
0answers
145 views

Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
4
votes
0answers
258 views

Is the universal constant in Caccioppoli's inequality one?

If you go through the classical proof of the fact that if $\Delta u= 0$ on $B_R$, then for every $R^\prime<R$ there holds $$ \int_{B_{R^\prime}}|\nabla u|^2 \leq ...
4
votes
0answers
89 views

Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...
4
votes
0answers
115 views

Carleman estimates on monotonicity formulas

I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from $$\int_{B_r} ...
4
votes
0answers
86 views

Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation \begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in ...
4
votes
0answers
81 views

Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
4
votes
0answers
163 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
4
votes
0answers
482 views

Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
4
votes
0answers
208 views

Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...
4
votes
0answers
120 views

Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question: Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$: $$ \frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t), $$ with smooth initial data ...
4
votes
0answers
193 views

Viscosity solution of the PDE

Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and $$|Du|-f(x,u)=0$$ where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
4
votes
0answers
329 views

Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...
4
votes
0answers
357 views

Existence of solutions to elliptic PDE in undbounded domain

Specifically, I have $Lu=f$, where $L$ is a linear elliptic pseudodifferential operator, on an unbounded domain of the form $\Omega\times [0,\infty)$, $\Omega$ has Lipschitz boundary, $u$ is 0 on all ...