0
votes
0answers
35 views

$L^\infty$ estimate for a fourth order (hyperbolic) equation

Consider the following fourth order equation $$u_{tt}+u_t= d\Delta u-\Delta^2u+f,$$ with Dirichlet or Navier boundary conditions, that is on $\partial\Omega$, we assume that ...
0
votes
1answer
138 views

When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...
0
votes
0answers
71 views

Estimate for an integral of a function of the solution to a PDE

Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and $\lambda\leq \sigma_1, \sigma_2 \leq ...
4
votes
2answers
170 views

Solution to Schrödinger equation

I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem. My goal is to solve this PDE for $f:[-1,1] ...
4
votes
2answers
146 views

Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form ...
9
votes
1answer
334 views

Special Second-Order PDE

Let $\Phi$ be a given smooth function on a neighborhood of zero in $\mathbb{R}^n$ with $$\Phi(0) = 0, ~~~~D \Phi(0) = 0, ~~~~ D^2\Phi(0) >0,$$ the latter meaning that the Hessian is positive ...
1
vote
0answers
68 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 ...
2
votes
2answers
116 views

Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?

Let us consider the basic linear elliptic PDE $$ \mathrm{div} (A\,\mathrm{grad}\,u) + bu = f, $$ with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$, $$ ...
1
vote
1answer
97 views

Existence of the solution of a linear parabolic pde

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in ...
0
votes
1answer
87 views

Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$

I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...
2
votes
1answer
155 views

Tempered distribution solution to a simple PDE

Let's consider the following PDE in $\mathbb R^d$ : $$\frac{\partial^d u}{\partial x_1...\partial x_d}=f$$ where $f$ is a tempered distribution with support in $\mathbb R^d_+$. There is a result by ...
4
votes
0answers
223 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 $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
0
votes
0answers
51 views

Why is it impossible to reduce a linear PDE of the second order in more than two independent variables to canonical form globally

It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential ...
2
votes
1answer
116 views

Extending a harmonic function in a ball to subharmonic in a larger ball

Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$: $$ \begin{cases} -\Delta u &= 0, \quad \text {in} \quad B(r), \\ \ \ \ \ \ \, u&= g, \quad \text {in}\quad ...
0
votes
0answers
104 views

Solvable PDEs and their Green's functions

I have a class of PDEs of the form $$ -\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0 $$ with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
1
vote
1answer
199 views

Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$

Consider the elliptic eigenvalue problem $$ \begin{cases} \int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\ \qquad \qquad \qquad \quad ...
2
votes
1answer
149 views

The centralizer of Lienard equation

Consider the lienard vector field $\cases{ x'=y -F(x) \\ y'=-x } $ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
3
votes
0answers
85 views

Linear heat equation with initial condition of generalized function

I am consider a very simple heat equation over the interval $[0, 1]$ with a Neumann BC and a very bad initial condition, written as: $\partial_tu(t, x) = \partial^2_xu(t, x) + a(t, x)u(t, x)$, for ...
1
vote
0answers
129 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 $$ ...
1
vote
0answers
72 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 ...
2
votes
1answer
178 views

Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$ Further assume a solvability condition $$\int_\Omega f ...
1
vote
2answers
98 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 ...
2
votes
2answers
115 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 ...
2
votes
1answer
247 views

Best approach to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
7
votes
2answers
246 views

Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
4
votes
1answer
165 views

Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
2
votes
1answer
267 views

Uniqueness of classical solution with degenerate boundary

Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$ in the form of $$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad (x,t) \in \Omega$$ with initial data $u(x,0) = x$ for ...
0
votes
1answer
155 views

Concerning Fritz John's article, The Ultrahyperbolic Differential Equation With Four Independent Variables

I am trying to read Fritz John's article, here: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1077490637 And for the proof of Thm 1.1 in page ...
2
votes
1answer
129 views

The maximum in the Poisson problem on the cube with constant source

Question: Let us consider the Poisson problem on the square with constant source $1$ $$ \begin{cases} - \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\ u &= 0, \qquad \text{ on } \partial ...
3
votes
0answers
219 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 ...
6
votes
2answers
259 views

Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
6
votes
0answers
325 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
1answer
450 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 ...
1
vote
1answer
154 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
votes
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 $\int f(x) dx =0$ . ...
0
votes
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
votes
1answer
859 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 ...
8
votes
3answers
3k views

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} ...
4
votes
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- $$u_\eta- v_\zeta= 0$$ ...
1
vote
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 ...
0
votes
1answer
525 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 ...
7
votes
0answers
312 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
votes
0answers
368 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 ...
1
vote
2answers
493 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 $v$ ...
8
votes
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 ...
0
votes
0answers
513 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 ...
2
votes
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
votes
1answer
576 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 $$ u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y) $$ ...
1
vote
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 ...
6
votes
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 ...