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

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1answer
147 views

Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?

We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it. Here ...
5
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1answer
449 views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
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1answer
153 views

A Cauchy problem for an iterated Euler-Poisson-Darboux eqaution

Good morning, I'm interested in solving a Cauchy problem for the iterated singular EPD. Well, Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the ...
3
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1answer
272 views

Upper bounds for the solution of an elliptic PDE depending on a parameter.

Suppose I have the following PDE on $[0,1]^n$ $$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$ with periodic boundary conditions and $\int f(x) dx =0$ . ...
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1answer
259 views

LINEAR Parabolic equations. Smooth dependence from initial data

I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data. More specifically I have the following problem: CONSIDER spaces $P:=\mathbb{R}^k$ ...
5
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1answer
858 views

Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
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0answers
387 views

What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?
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0answers
114 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 ...
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1answer
146 views

Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients. $$ f = \sum_I f_I \mathbf x^I $$ This power series is said to ...
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1answer
282 views

Linear inhomogeneous PDE

I am trying to understand the behavior of the following linear PDE: $$\partial_t u(x,t) = \partial_{xx} u(x,t) + f(x) u(x,t)$$ where I set $f(x)=\lambda e^{-x^2} -1$ and with: IC : $u(x,0)= 1$ on ...
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3answers
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Physical Interpretation of Robin Boundary Conditions

In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet $u|_{\partial\Omega} \equiv 0$, Neumann $D_{\nu} ...
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2answers
205 views

hodographic transformation

Let $\phi(x,t)$ be smooth function. Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$. Let $u$, $v$ satisfies following equations: 1- $$u_\eta- v_\zeta= 0$$ ...
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0answers
136 views

coercivity vs boundedness of operator

hi, the definition of coercivity and boundedness of a linear operator L between two B spaces looks similar: $\|Lx\|\ge M _ 1\|x\|$ and $||Lx||\le M _ 2\|x\|$ for some constants $M _ 1$ and $M _ 2$. ...
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0answers
173 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 ...
3
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1answer
455 views

Stochastic Heat Equation

Given the heat equation: $$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$ with the boundary conditions: $$\Phi(x,0)=\Phi_0$$ and a Neumann boundary condition of the kind: ...
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1answer
524 views

Hopf Boundary Point Lemma

The Hopf Boundary Point Lemma http://en.wikipedia.org/wiki/Hopf_lemma is a result for the unit normal vector field and the normal derivative. Is it true if one considers arbitrary directional ...
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311 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 ...
3
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0answers
367 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 ...
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2answers
492 views

Heat equation with Neumann BC

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$. Is this true to say: $$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ ...
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9answers
3k views

Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...
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0answers
512 views

characteristic surface

Hello, I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ : (1) $G_{xy}=0$ (2) $G_{xz}=0$ (3) $G_{yz}=0$ (4) $G_{xx}-G_{yy}=0$. It is not hard to see that the general ...
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0answers
259 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 ...
3
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1answer
575 views

Long time behavior of the heat equation on R

Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is $$ u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y) $$ ...
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3answers
440 views

How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets

$Qn#1 $ : Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...
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2answers
254 views

Structure of solutions of a PDE from a game theory problem

I found the following the following differential equation in the context of a Game Theory problem. I was wondering if this is related to any known family of equations or whether there is any hint ...
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1answer
374 views

orthonormal basis of eigenvectors for laplacian on a concave polygon

I am interested in the Laplace operator $\Delta$ on a concave polygon. When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$ is boundedly invertible. In addition, ...
6
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3answers
756 views

Analytical solution to a Linear advection-reaction PDE

I am looking for an analytical solution for the linear PDE $(1)\qquad\qquad \qquad f_t+ A f_x + B f = 0, $ Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector. Clearly each one of ...