# Tagged Questions

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### Di Perna-Lions theory for transport equation

Does someone know if some notes on the topic mentioned in the title are available online? I'm reading the paper "Ordinary differential equations, transport theory and Sobolev spaces" by Di Perna and ...
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### Existence and Uniqueness of Solutions of PDEs

I have been looking into the Cauchy-Kovalevskaya Theorem where one can "establish the local existence of analytic solutions to a system of PDEs". I wanted to see an application (for example, see ...
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### Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
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### Third Order PDE written as a System of (Linear) First Order PDEs

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs. I have no idea how to tackle this problem. Any form of help will be appreciated. ...
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### Maximum principle for linear elliptic operators of arbitrary order

What is known about maximum principles for strongly elliptic linear differential operators of even order (possibly higher than $2$)? By such an operator, I mean a linear differential operator with ...
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### Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
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### References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
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### Characterization of weak solution

5 Nonlinear elliptic variational inequalities Preliminaries In order to explain the importance of elliptic variational inequalities, first consider the weak solution of the linear ...
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### System of ODEs and DAE system

Let us consider the following system of ODEs: $$y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0$$ and the following one: $$y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0.$$ $f$ and $g$ ...
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### Regularity of a Weak Solution

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
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### Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$\Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
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### Perron solution and weak solution for a Dirichlet problem in a convex domain

Consider $\Omega \subset R^n$ an open, bounded and convex set. Then your boundary is Lipschtz. Then we can define the trace operator T. Consider $K \subset \partial \Omega$ a compact set with non ...
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### about the Perron method for the Dirichlet problem

Consider $\Omega$ an open, convex and bounded set of $R^n$. Let $g: \partial \Omega \rightarrow R$ a function. Supose that $g$ is continuous except in one point. By the convexity of $\Omega$ we can ...
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### Why has no body retypeset Ladyzhenskaya et al's “Linear and quasi-linear equations of parabolic type”? [closed]

The book "Linear and quasi-linear equations of parabolic type" is one of the ugliest books I have ever seen in my life. The fonts are awful, the notation is difficult to understand and recall and the ...
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### When does weakly elliptic $\Rightarrow$ strongly elliptic?

While learning more about the analytic background for the Atiyah-Singer Index Theorem, I was curious about the following question (although not needed for the ASID): what are some general conditions ...
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### Elliptic equation on a cylinder with mixed Dirichlet-Robin conditions

For an elliptic equation on a finite 2-dimensional cylinder, with homogeneous Dirichlet boundary conditions at the bottom and Robin conditions on the top, does there exist elliptic estimates on the ...
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### Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
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### Methods of characteristic for system of first order linear hyperbolic partial differential equations: reference and examples

I would like to understand a few points on the methods of characteristics used to resolve a system of coupled, linear first order partial differential equation (of the hyperbolic type). Some example ...
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### Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
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### Characterization of Sobolev Space

I have just started learning about Sobolev spaces. So this might be trivial. I am working through the book "Partial Differential Equations" by Lawrence Evans, it came highly recommended. Taking ...
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### generalization of mean value property of subsolutions

proof of an generalization of the mean value property If $\mathcal{L}u \ge 0$ in $B_{2\rho}$, then \sup_{B_\rho} \le C \left (\dfrac{1}{|B_{2\rho}|}\int_{B_{2 \rho}} |u|^p dx \right ...
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### Reference on $\mathcal{L}^p(I;X)$

I am doing some reading on evolution equations, and $\mathcal{L}^p$ spaces with functions with values in a Banach space $X$ appears rather often. However I have not found a comprehensive reference ...
### $H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required
Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...