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

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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_{\...
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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,...
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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 ...
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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 ...
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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\...
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1answer
112 views

Properties of connection Laplacian on vector fields

Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...
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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{\...
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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 $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$$...
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1answer
115 views

$L^\infty$ estimate on heat equation with a lower order term

Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ $$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$ ...
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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 $n$ ...
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1answer
125 views

Harmonic/Subharmonic lifting of functions on an annulus

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...
2
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1answer
81 views

How to find the eigenvalues equation of this PDE problem

Given the problem: $$(\kappa(x)X^{'})^{'}+\lambda\rho(x)X=0$$ for $0<x<l$ with $X(0)=X(l)=0$ where $\kappa(x)=\kappa_{1}^{2}$ for $x<a$, $\kappa(x)=\kappa_{2}^{2}$ for $\kappa>a$. $\rho(x)=...
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1answer
64 views

Why are the tangential derivatives in this diffraction problem zero? [closed]

I'm considering the diffraction problem described in section 3.16 of "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Uraltseva (1968). Let $\Omega$ be an open bounded subset in $\...
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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 ...
2
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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 \...
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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_{\...
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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 ...
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2answers
86 views

Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs

I have the following system of PDEs that I want to solve as "analytically" as possible: $$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$ where $A$ and $B$ are constant, ...
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2answers
310 views

Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad ...
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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 \,...
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22 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 ...
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2answers
198 views

Airy's equation on $\mathbb R_-$

I am interested in Airy's equation $$\frac{\partial u}{\partial t}(t,x)=-\frac{\partial^3 u}{\partial x^3}(t,x)$$ on a bounded or semi-bounded domain, e.g. on $(-\infty,0)$. In order to obtain a group ...
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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 $...
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2answers
144 views

A Global Estimates for Linear Elliptic PDE

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy $-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$, where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
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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}...
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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 ...
3
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0answers
79 views

Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :) Consider the following Laplace boundary value problem (BVP) $$\matrix{ {{\nabla ^2}\Phi (x,y)...
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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 ...
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1answer
98 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 ...
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1answer
97 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 ...
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1answer
269 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 \\ &...
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1answer
227 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 ...
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3answers
165 views

Does “solutions of an $n$-th order ODE form an $n$-dimensional vector space” somehow generalise to PDEs?

It is well known that the set of solutions $u:\mathbb{R}\rightarrow \mathbb{R}$ of an $n$-th order, linear, homogeneous ordinary differential equation $$a_n(x)\frac{d^n u}{dx^n}+\dots + a_1(x)\frac{du}...
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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{...
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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=(...
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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 $...
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2answers
322 views

Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions. Unfortunately, I am a ...
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2answers
209 views

Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science. I am interested in proving or obtaining a counterexample to the following conjecture. Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...
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1answer
293 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
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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 , ...
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1answer
89 views

The hypoellipticity of a heat-like operator

I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac {...
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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 ...
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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+...
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1answer
177 views

Elliptic pde with bilaplacian; boundary conditions.

I am interested in the solvability of $$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ \...
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2answers
102 views

Separation of variables for a particular PDE

Given the partial differential equation \begin{equation} (1-x)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial x} \right] + (1-y)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial y} \right] = 0 \...
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2answers
243 views

Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...
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1answer
214 views

Gradient estimate for elliptic equation

Given: 1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$ 2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$ 3)$g=(g_i,\ldots,g_n)\in \mathcal{C}^\alpha(\...
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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 ...
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1answer
145 views

elliptic boundary regularity, tangential regularity

A have a question related to the boundary regularity of a solution of a Poisson equation on a bounded domain. But to make the question easier to pose I will state it on $ R_+^2:=\{ x \in R^2:x_2>0\...
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2answers
253 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 ...