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

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7
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0answers
309 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 ...
6
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0answers
324 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 ...
3
votes
0answers
47 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) + ...
3
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0answers
210 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 $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
3
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0answers
84 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 ...
3
votes
0answers
215 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
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 ...
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 ...
2
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0answers
256 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 ...
1
vote
0answers
66 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
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0answers
127 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
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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 ...
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
0answers
20 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
0answers
31 views

existence of solution of volterra integral equation for the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
votes
0answers
69 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 ...
0
votes
0answers
98 views

positive eigenfunction on complete Riemannian manifold

Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation $$ \Delta u = u $$ where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...
0
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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 ...
0
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0answers
51 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 ...
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 ...
0
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0answers
385 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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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 $M _ 2$. ...
0
votes
0answers
508 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 ...