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

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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 ...
1
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0answers
78 views

Elliptic problem on half space; infinite boundary values; Liouville theorem

In a the study of a boundary value problem the following problem is arising: $-\Delta v(x)= e^{v(x)}$ in $ R^N_+$ $v= - \infty$ $\qquad $ on $ \partial R^N_+$ $ \qquad $ $ v \le 0$ in $ R^N_+$. ...
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votes
1answer
108 views

Application of Toms- Stein restriction theorem for Strichartz estimates

The initial value problem for one dimensional Shrödinger equation is $$iu_{t}+u_{xx}=0,$$ $$u(x, 0)= f(x),$$ where $u:\mathbb R \times \mathbb R \rightarrow \mathbb C$ is a complex valued ...
3
votes
0answers
188 views

Uniqueness theorems related to Hardy Uncertainty Principle

Uncertainty Principles state that a function and its Fourier transform cannot be simultaneously sharply localised. A well known result due to G.H.Hardy says that if $f(x)=O(e^{-\alpha^2|x|^2})$, ...
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1answer
124 views

Proof of global (in time) existence of classical solutions for 2D Euler equation in bounded domain

Anyone can explain the main idea, or recommend some paper or book on that? For the whole space case, or the periodic case, the proofs are everywhere. But those do not seem to apply to the bounded ...
2
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1answer
222 views

unique continuation property for overdetermined elliptic PDE

On a closed manifold $M$, let $P(f)=0$ be a linear overdetermined elliptic system of PDE of 2nd order with smooth coefficients. By overdetermined ellipticity, I mean the principal sympbol is ...
1
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1answer
205 views

Reference request: Spectral analysis of advection diffusion PDE

As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and NOT on existence/uniqueness etc. which is usually the ...
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1answer
103 views

analytic solution to elliptic PDE in R^n

I am looking for (minimal) conditions, which guarantee that the problem Lu = 0 in R^n, where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global ...
3
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0answers
96 views

Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition. What can ...
3
votes
1answer
227 views

A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation

While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two ...
2
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0answers
81 views

Elliptic equations with divergence-free drift terms

Given $\ \mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ bounded, $div$$(\mathbf{u})=0$, ...
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2answers
254 views

Reference Request: Probability and (Nonlinear) PDEs

I'm a graduate student interested in learning about probability and (mostly evolutionary) PDEs, just for fun (and as an excuse to learn some probability). I'm mostly interested in things along the ...
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2answers
306 views

Reference request: learn measure theory for PDEs

I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...
1
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2answers
134 views

Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data

I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial ...
2
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2answers
112 views

Is the left regularizer for elliptic BVP a left inverse for the principal part?

Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
1
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0answers
174 views

local existence for a singular quasilinear parabolic equation

I'm considering the following type of PDE: $u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$ with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, ...
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0answers
62 views

Kernel of perturbation of biharmonic operator

Suppose we have a linear fourth order operator defined on $\mathbb{R}^{2n}$ with $n\geq2$ of the form: $$\mathcal{L}(f)=\Delta^{2}f+\sum_{i,j=1}^{2n}P_{ij}(x)\partial_{i}\partial_{j}f$$ with ...
3
votes
0answers
93 views

Critical elliptic equation; kernel of linearized operator

I am interested in the critical equation $- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin ...
1
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1answer
177 views

Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful. Let $V \subset H ...
4
votes
0answers
96 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 ...
3
votes
1answer
198 views

Two equivalent definitions of weak solution to parabolic PDE; don't understand proof

(Crossposted from MSE due to no replies) I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the ...
3
votes
1answer
192 views

Heat Equation on $[0,T] \times \mathbb{R}^n$

I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain. Do you ...
0
votes
1answer
82 views

approximating functions pointwise [closed]

If we have in certain norm 1). $g_j(x) \rightarrow h(x), j\rightarrow\infty$ and 2). $ f_{ij}(x)\rightarrow g_j(x), i\rightarrow \infty, \forall j$ , then we can choose a subsequence ...
5
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3answers
359 views

PDEs involving measures; where to begin?

If I want to learn about existence of weak solutions to PDEs of the form $$u_t + Au = f$$ or $$Au = f$$ where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for ...
2
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0answers
72 views

Helmhotz decomposition and Regularity in Stokes equation

It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely decomposed as \begin{eqnarray*} \ f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)} \ \end{eqnarray*} with $f_{0}\in ...
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4answers
428 views

Einstein field equations in perspectives from PDE and functional analysis

The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
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0answers
81 views

Interpretation of this PDE solving method needed

I am working on an open source project that deals with the usage of lie groups to solve first order differential equations. For my reference, I am using this paper by Maple (It is there on this ...
3
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0answers
129 views

How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded

With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider $$u' + Au = f$$ in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
3
votes
0answers
140 views

Tangential boundary regularity for optimal transport maps

I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II: Assume $u$ is a convex ...
2
votes
2answers
248 views

worst regularity of f ensuring u is locally Lipschitz for $\Delta u = f$

Assume $M$ is a Riemannian manifold, and $\Omega $ is a bounded domain. Consider the Poisson equation: $$\Delta u = f \qquad \text{with }u \in {W^{1,2}}$$ What is the worst regularity of $f$ which ...
3
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1answer
156 views

A closed extension of the Laplace operator with respect to the supremum norm

Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
10
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0answers
335 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 $K$. ...
3
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0answers
112 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
3
votes
2answers
232 views

Non symmetric coefficient matrix for elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form $$ D_i(a_{i,j}D_ju)=0 \tag{1}$$ where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
0
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0answers
49 views

partial maximum principle for elliptic differential operators

Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...
5
votes
1answer
461 views

Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one. $I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...
3
votes
1answer
167 views

Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
3
votes
1answer
97 views

On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument

Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...
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0answers
69 views

Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?

Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces. I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that $$\langle u'(t), ...
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0answers
112 views

Galerkin approximations for parabolic PDE weak solution, getting a uniform bound

(As usual $V \subset H$ are separable Hilbert spaces) In a book I read this about existence of the solutions to parabolic PDEs: the approximate solution $u_n(t)$ solves the equation $$(u_n', ...
2
votes
1answer
89 views

Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?

Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow $$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = ...
2
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0answers
58 views

Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces

Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
2
votes
0answers
109 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
2
votes
1answer
239 views

Best approach to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
2
votes
0answers
106 views

linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator: $$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$ where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
1
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1answer
104 views

metric scaling for an inequality

I read a lemma 1.12 in Tobias H.Colding's paper "Ricci curvature and volume convergence"."Suppose that $Ri{c_{{M^n}}} \ge \left( {n - 1} \right)\Lambda {R^{ - 2}}$,p and $q \in M$ with d(p,q)>8R,and ...
1
vote
0answers
263 views

A variation of Poisson's equation in cylindrical coordinates

Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus: $\nabla^2 V = \frac{f(V)}{R^2}$ ...
2
votes
0answers
91 views

Regularity of solution of nonlinear equation

Hi! Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball $B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$. Let $f\in C^{0,\alpha}(B_1)$ ...
0
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0answers
133 views

Definitions of weak solution to parabolic PDE

Suppose $$u'(t) + A(t)u = f(t)$$ holds in the sense of $L^2(0,T;V')$. Is the problem 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), ...
1
vote
1answer
122 views

Reference for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$

If it is true, where may I find a reference/proof for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$ where $H$ is a Hilbert space. Thanks