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Examples of subharmonic functions

Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
Adi's user avatar
  • 455
0 votes
0 answers
17 views

Third order estimate for linear elliptic equations

Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
Adi's user avatar
  • 455
0 votes
0 answers
87 views

Curl-Div equation with singular matrix

I want to solve the equation: $$ \begin{cases} \nabla \times (A \mathbf v)=f, \quad x\in \Omega \\ \operatorname{div}(\mathbf v)=0, \end{cases} $$ where $\Omega \subset\mathbb{R}^n$, is an open set, $...
Gustave's user avatar
  • 617
11 votes
0 answers
317 views
+50

Sobolev's PDE Scottish Book Problem (Problem 188)

In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution. In 2015, when the second edition of the Scottish Book with updates and commentary on ...
Mark Lewko's user avatar
1 vote
0 answers
23 views

Uniform bound on the first moment for a perturbed advection-diffusion equation

I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line: $$ \begin{cases} u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x = ...
Garou Garou's user avatar
1 vote
0 answers
39 views

Hyperbolic equation without initial state

Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$: $$ a^2 u_{xx} - b^2 u_{yy} = f(x, y), $$ with Dirichlet boundary conditions on $u$. By using the ...
Gustave's user avatar
  • 617
3 votes
1 answer
212 views

$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$

I want to solve the equation: $$ \begin{cases} \nabla \times (F\times\mathbf v)=g, \\ \operatorname{div}(\mathbf v)=0, \end{cases}\label{1}\tag{1} $$ where $F$ and $g$ are given vector fields. The ...
Gustave's user avatar
  • 617
5 votes
0 answers
112 views

Fredholm index of degenerate elliptic PDE

We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$: $$ L(u):= -\operatorname{div}(a \, \nabla u) = 0, $$ where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$...
Dorian's user avatar
  • 363
4 votes
0 answers
92 views

Well-posedness for linear transport equations with fractional diffusion term

I have a rather applied problem where I consider an equation of the form $$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$ ...
confused postdoc's user avatar
6 votes
2 answers
250 views

Minimal assumptions for existence of solutions of First order PDE

I'm looking for a reference about existence of linear homogeneous first order PDE, in particular about the minimal assumption on the data. In literature I found that one require $C^1$-regularity on ...
Simmetrico's user avatar
9 votes
2 answers
493 views

Reference Request for global Hölder continuity of solutions to elliptic PDEs

This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
DarkViole7's user avatar
4 votes
1 answer
308 views

A certain solution for Sine-Gordon Equation

I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
RWien's user avatar
  • 245
5 votes
2 answers
267 views

Positivity for the mild solution of a heat equation on the torus

Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
kumquat's user avatar
  • 185
0 votes
0 answers
37 views

Finding a particular solution to a Lax pair on $[0, \infty)$ to solve $q_t + q_{xxx} + u(x)q = 0$ using the Fokas Method

I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here). This post is a follow up to my first that ...
Talmsmen's user avatar
  • 547
0 votes
0 answers
40 views

Multivariable Variation of Parameters $\psi_1(x, k)\int^{(x,t)}e^{ik^3(t-\tau)}[M_1(\xi,k)q(\xi,\tau)d\xi-X_1(\xi,\tau,k)d\tau]+\dots$

I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here). For greater clarity, I split my question up into two ...
Talmsmen's user avatar
  • 547
3 votes
1 answer
86 views

$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line

I consider the following problem on the half real line $$ \begin{cases} u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm] u|...
Garou Garou's user avatar
1 vote
0 answers
120 views

Well-posedness result for a linear parabolic equation on the torus

Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
kumquat's user avatar
  • 185
0 votes
0 answers
69 views

Inside and up to boundary regularity improvement of linear differential operator

I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$ $$A=\sum a_\alpha\...
Holden Lyu's user avatar
0 votes
0 answers
46 views

Uniqueness results for linear first order systems of PDEs

Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$): $$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
Samuele's user avatar
  • 1,205
2 votes
0 answers
238 views

What is the fundamental solution for the backward heat equation?

According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
Andrew's user avatar
  • 2,715
0 votes
0 answers
62 views

Uniqueness problem of constant coefficient differential operator with given boundary information on compact domain

I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have \begin{equation}\label{...
Holden Lyu's user avatar
1 vote
0 answers
40 views

Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?

Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$: \begin{equation} u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1} \end{equation} Assume that $...
Guohuan Zhao's user avatar
4 votes
3 answers
473 views

Generalized Fuchsian-type PDE

Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
Math2024's user avatar
  • 141
0 votes
1 answer
217 views

About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain

I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$: Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
Stack_Underflow's user avatar
7 votes
2 answers
567 views

Intuition for Agmon-Douglis-Nirenberg ellipticity

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently. I am trying to understand the definition of ellipticity of systems due to ...
G. Blaickner's user avatar
  • 1,429
0 votes
0 answers
111 views

Reversing heat transfer with respect to time

Fact: One can easily compute heat dispersion in a plane using the heat equation. Question: Has any research been done on computing the process in the reverse time direction? That is, given a heat map $...
Snared's user avatar
  • 119
1 vote
0 answers
211 views

Understanding the effect of PDE solution on critical strip?

I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
John McManus's user avatar
3 votes
0 answers
65 views

Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
Paruru's user avatar
  • 51
1 vote
0 answers
131 views

Regularity of elliptic equation with Neumann boundary conditions

In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
Cathelion's user avatar
1 vote
0 answers
32 views

Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation

I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO). Consider the following initial boundary value problem for the linear ...
user111's user avatar
  • 4,034
1 vote
1 answer
149 views

Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data

I already asked the question on MSE, and have tried to figure it out myself. But the problem seems trickier than expected, so I guess MO is a better place to ask.. For the sake of completeness, I ...
Isaac's user avatar
  • 3,477
1 vote
0 answers
106 views

Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
SebastianP's user avatar
2 votes
0 answers
153 views

Uniqueness of the solution to systems of first-order linear PDEs

Context: Let $\Omega \subset \mathbb{R}^p$ be an domain. For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
Paruru's user avatar
  • 51
4 votes
2 answers
283 views

Regularity of solution of $(-\Delta + w)f = 0$

I am studying the following Schrödinger equation: $$(-\Delta + w)f = 0$$ which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
MathMath's user avatar
  • 1,305
5 votes
1 answer
238 views

$L^2$ regularity theory for elliptic equations: Is there another method other that the difference quotient method? Reference request

So i'm interested in the following classical theorem or similar variants. Consider the following elliptic PDE $$ -D_\alpha(a^{ij}D_\beta u) = f. $$ If we assume that the coefficients $a^{ij}$ are ...
Franlezana's user avatar
2 votes
1 answer
391 views

Estimates for linear elliptic PDE in the whole of $\mathbb{R}^n$

I am struggling to track down any literature on the topic of elliptic regularity when the domain in question is the whole space $\mathbb{R}^n$. Consider the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^n)$,...
Benjamin's user avatar
4 votes
2 answers
781 views

Is there any bilinear Poincaré/Sobolev inequality?

Is the following, I call it bilinear Poincaré inequality, true? Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
Hao Yu's user avatar
  • 185
3 votes
0 answers
62 views

Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field

Setting Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
MyShepherd's user avatar
1 vote
0 answers
38 views

Studying the evolution of laplacian in NS equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by: \begin{equation}\label{Eq1} \dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
MrPie 's user avatar
  • 317
1 vote
0 answers
42 views

Mixed boundary condition of parabolic equations

Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that $$ \partial\Omega=\partial\Omega_D\cup\partial\Omega_N, $$ where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
Luis Yanka Annalisc's user avatar
4 votes
1 answer
168 views

Continuous up to the boundary without boundary smoothness

Let $\Omega$ be a bounded domain, $f\in L^{\infty}(\Omega),$ and $0\leq u\in H_0^1(\Omega)$ is a non-negative solution of $\Delta u=f$. My question is as follows: Can we conclude that $u\in C^0(\bar{\...
sorrymaker's user avatar
2 votes
1 answer
172 views

Method of characteristics with 2 dependent variables in 3 dimensions

I'm interested in solving a first-order linear PDE with 2 dependent variables in 3 dimensions by the method of characteristics. Something of this general form: $$ A \frac{\partial u}{\partial x} + B \...
Computerish's user avatar
2 votes
0 answers
60 views

Dirichlet's problem for Laplace's equation in the model domain

Let $$\Omega_\alpha=\left\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\right\},$$ $a>0$. we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{ w}_0^...
sidi mohamd deval's user avatar
2 votes
0 answers
183 views

Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply. When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
monotone operator's user avatar
5 votes
0 answers
417 views

All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)

I have asked this question on MSE, but this is a better place. The heat equation and the heat kernel. Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
Lorenzo Pompili's user avatar
4 votes
0 answers
111 views

Continuity of solutions of Elliptic PDE with respect to parameters

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy $$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$ where $f$ is a fast decaying smooth function. I would like to know how the ...
A random mathematician's user avatar
3 votes
0 answers
103 views

Comparison principle for Elliptic PDE with exponential nonlinearity

Suppose $\varphi$ is a radial (and radially decreasing) solution of $$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$ with $ R>0$, and $\psi$ is a decreasing radial function ...
A random mathematician's user avatar
2 votes
0 answers
114 views

A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define $$ H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N. $$ By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
Lucas Linhares's user avatar
1 vote
1 answer
112 views

Solving a particular delay PDE $\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s)$

I recently encountered a particular delay PDE in my work, the solution of which corresponds to the Laplace transform of some probability distribution. I'm having trouble to solve this equation. The ...
ely's user avatar
  • 13
1 vote
0 answers
315 views

Maximal regularity heat equation

Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate \begin{align*} \forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
Ayman Moussa's user avatar
  • 3,425

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