Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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9
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0answers
230 views

Does hypoellipticity imply the existence of a parametrix?

Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...
7
votes
0answers
372 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 $u:...
6
votes
0answers
362 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 ...
5
votes
0answers
519 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 $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
4
votes
0answers
331 views

well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
3
votes
0answers
40 views

Global Euclidean Carleman Estimate with a linear phase

I am interested in deriving the following global Carleman estimate which I think should hold : $ \| e^{\tau \phi} \triangle u \|_{L_{\delta}^2({\mathbb{R^3})}}> C \tau \| e^{\tau \phi} u \|_{L^2_{\...
3
votes
0answers
223 views

Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
3
votes
0answers
89 views

How can one do change of variables for solutions to a staochastic partial differential equation?

isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation: $$ dy(t,x) + y_x(t,x) + y(t,...
3
votes
0answers
131 views

Quadratic forms over symmetric $3\times3$ matrices

This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...
3
votes
0answers
388 views

Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference: Let us consider the orientation-preserving homeomorphic solutions $f: D \...
2
votes
0answers
56 views

Fractional derivative of the Wright function

It is mentioned in some papers (Appendix in this paper, for example) that the (formal) solution of the fractional drift (or transport) equation $$ \partial_{t}^{\alpha}u(t,x)+\partial_{x}u(t,x)=0\quad\...
2
votes
0answers
50 views

Constant in a trace Sobolev theorem for concave domains

I wonder is the following inequality is true/known: Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then $$ \int_{\partial\Omega} |u|^2 ds \...
2
votes
0answers
44 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) \mbox{...
2
votes
0answers
79 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 $ \Omega=(...
2
votes
0answers
55 views

Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
2
votes
0answers
53 views

only solution to wave equation under certain restriction?

Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two bounded solutions to the wave equation \begin{equation} \partial_t^2u_i = \partial_x^2u_i \end{equation} obeying the restriction $\...
2
votes
0answers
81 views

Solve a PDE related to free boundary problem

I would like to solve the following system for my problem: $$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$ where $u=u(s,l): R\times R_+\to R$ is the unknown function ...
2
votes
0answers
291 views

General solutions for HJB equations in a special case.

I am reading the book of Wendell Flemming in control theorem to learn the HJB equation Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that $0\...
1
vote
0answers
61 views

Method of characteristic for a system of first order PDEs

I am working with this system of first order PDEs: \begin{equation} \left\{ \begin{aligned} %Suscettibili &\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
1
vote
0answers
73 views

$C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$$...
1
vote
0answers
36 views

About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines". For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
1
vote
0answers
33 views

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too. Consider the following ...
1
vote
0answers
40 views

Fundamental gap for Neumann BVP with potential

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem \begin{cases}...
1
vote
0answers
61 views

Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity). Let $ \Omega$ denote a cube in $ R^n$ and consider ...
1
vote
0answers
27 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
85 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 $...
1
vote
0answers
73 views

biharmonic equation with L^1 data and Navier Condition

I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
1
vote
0answers
145 views

Gilbarg-Trudinger's book Theorem 4.13

I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13. Theorem 4.13 is a special case of Kellogg's theorem in ...
1
vote
0answers
29 views

Renormalization for Transport Equations with SBD velocity field

In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in SBD(...
1
vote
0answers
114 views

Existence of Green's functions for PDEs

Here is what I think I know: Given a symmetric linear differential operator $\mathcal{L}$ that is positive definite on a function space(/space of distributions) $\mathcal{H}$, we can find its inverse, ...
1
vote
0answers
150 views

Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here. Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let $\mathcal{...
1
vote
0answers
91 views

Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$. Consider the system, with $u^\epsilon, v^\epsilon \...
1
vote
0answers
70 views

References Request : Existence and Uniqueness for PDE which is “ALMOST (?)” Parabolic

I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case: \begin{equation} Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + \sum_{...
1
vote
0answers
111 views

A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple. There is a linear operator $L:{D}(L) ...
1
vote
0answers
92 views

Limit Toward Discontinuous Point of Dirichlet Boundary Value

The question arises from a paper on Schwarz's domain decomposition method (click here). We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below. Now we ...
1
vote
0answers
74 views

Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
1
vote
0answers
280 views

Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations singular at some point? For example, I am analyzing a partial differential equation $$ \partial_{t}u=\partial_{rr}u+\frac{2}{r}\...
1
vote
0answers
82 views

Analyticity of one-dimensional PDE solutions with respect to the space variable

Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients $$ u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0, $$ in some neighborhood of a point $(x_0,...
1
vote
0answers
217 views

Is this Stefan-type problem an open problem?

I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
0
votes
0answers
19 views

Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy, $$ \int_{\Omega}|\nabla u |^2\, -\, \int_{\...
0
votes
0answers
63 views

Fundamental solution matrix of a linear PDE

I've asked a very similar question also at math.stackexchange, but I've not received any answer. A vectorial function $\boldsymbol{x}:\mathbb{R}^D \rightarrow \mathbb{R}^N$ satisfies the following ...
0
votes
0answers
41 views

When can an analytical solution for the heat equation be obtained?

I am currently trying to model a system with a time varying heat flux. It seems most researchers are using FEM to obtain the heat distribution (solve the heat equation). When can the heat equation be ...
0
votes
0answers
50 views

High dimensional partial differential equation

I encountered the following partial equation. Let $f(z,x_1,\cdots,x_n)$ be a function with $n+1$ entries.Let $a_i,b,c$ be constants. $$ \sum_{i=1}^n \frac{a_i}{(x_i-z)^2}+\frac{b}{z(z+1)}-\frac{\...
0
votes
0answers
27 views

Solving a system of PDEs

I want to solve a system of partial differential equations of the form $\sum_{kl} A_{ikl}\frac{\partial u_{kj}}{\partial x_l}+\sum_k B_{ik} x_k u_{kj}=c_{ij}$ where $u_{ij}(x_1,...,x_N)$ are the ...
0
votes
0answers
22 views

Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D: $$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g \,...
0
votes
0answers
23 views

Boundary Conditions for Linear PDEs using “Lagrange Multipliers”

Is there a way to take a linear PDE with homogeneous boundary conditions and somehow change the differential operator to include a sort of "Lagrange Multiplier" and then have a more complex (not ...
0
votes
0answers
108 views

For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?

Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that $...
0
votes
0answers
76 views

solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
0
votes
0answers
157 views

Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as: \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0 \end{equation*} where ...
0
votes
0answers
66 views

Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...