1
vote
0answers
25 views

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 ...
1
vote
0answers
18 views

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 ...
0
votes
0answers
53 views
+50

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, ...
2
votes
1answer
35 views

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. ...
0
votes
0answers
20 views

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 ...
2
votes
0answers
43 views

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? ...
2
votes
1answer
35 views

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 ...
3
votes
1answer
59 views

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 ...
1
vote
0answers
21 views

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 $g$ ...
3
votes
0answers
41 views

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 ...
1
vote
2answers
66 views

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 ...
0
votes
0answers
39 views

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 ...
1
vote
0answers
15 views

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 ...
2
votes
0answers
48 views

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 ...
2
votes
0answers
24 views

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 ...
0
votes
0answers
22 views

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 ...
0
votes
2answers
43 views

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 ...
0
votes
0answers
33 views

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 ...
35
votes
5answers
1k views

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 ...
2
votes
1answer
62 views

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 ...
2
votes
0answers
28 views

Which came first, energy minimization or pde?

I'm interested in a historical perspective on pde. I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ ...
3
votes
1answer
57 views

Elliptic PDEs in Banach space

The standard textbooks discuss weak solutions and regularities in Hilbert spaces $W^{k,2}.$ I could not find a good reference on the theory based on Banach spaces $W^{k,p}.$ It would be good to point ...
0
votes
1answer
21 views

Existence and regularity of 2D stream function.

Suppose $\Omega$ is a bounded open simply connected region in $\mathbb{R}^2$, and let $u:\overline{\Omega} \to \mathbb{R}^2$. Suppose $u$ is divergence free. Then there exists a stream function $\psi$ ...
0
votes
0answers
17 views

reference for existence and blow up results in transport-like PDEs (Vlasov equation)

I'm looking for references to results regarding maximal time existence of solutions of a certain transport-like PDE, more precisely this one: $$ \partial_t f + v \cdot \nabla_x f + E(f,t,x) \cdot ...
0
votes
0answers
7 views

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 \begin{equation} \sup_{B_\rho} \le C \left (\dfrac{1}{|B_{2\rho}|}\int_{B_{2 \rho}} |u|^p dx \right ...
1
vote
0answers
14 views

A Dirichlet problem with Landesman Lazer condition

In my book in the chapter about the saddle point theorem there has the following exercise: Let $\Omega \subset R^n$ $(N \geq 1)$ be a bounded , smooth domain and consider the Dirichlet problem $$ ...
8
votes
3answers
150 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
0
votes
0answers
13 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
2
votes
0answers
27 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
0
votes
1answer
36 views

What are good resources for learning Numerical methods for Partial Differential Equations?

I'm having an undergraduate course on Numerical Solutions to Ordinary and Partial Differential Equations. I need online resources to supplement my study preferably videos and books. I want to build a ...
0
votes
0answers
34 views

Elliptic regularity of Dirichlet problem

Suppose $\Omega\subset\mathbb{C}$ is a simply connected domain with $C^\infty$ boundary. Consider the following Dirichlet problem $$\Delta u |_{\Omega}=0 $$ $$u|_{\partial\Omega}=f$$ Under what ...
1
vote
0answers
44 views

Regularity of Dirichlet Eigenvalues on Lipschitz Domain

What kind of regularity do we generally have for weak solutions to the Dirichlet problem? $$(\Delta+\lambda)u=0 \textrm{ in }U$$ $$u=0 \textrm{ on }\partial U $$ where $U$ is a planar domain with ...
1
vote
0answers
21 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
1
vote
0answers
27 views

Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
0
votes
0answers
14 views

Classification of second order PDE

I am trying to understand the classification of second order PDE's from this article. In page 45, line 1 can somebody please explain to me how $\partial^2U \over {\partial x_i\partial x_j} $ was ...
0
votes
0answers
24 views

Reference for elliptic regularity for $-\triangle \phi + u \cdot \nabla\phi=f$ under minimal assumptions

I have a distributional solution to $-\triangle \phi + u \cdot \nabla \phi= f$ in $U \subseteq \mathbb{R}^n$ and $\phi=0$ on $\partial U$. I have that $U$ is open, bounded, connected, ...
2
votes
1answer
25 views

how to prove that this weak solution is subharmonic?

My question is about this article http://hal.inria.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf. My question is : Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = ...
4
votes
0answers
78 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
0
votes
1answer
25 views

Reference needed for: $u \in H^1(0,T;L^2)$ if and only if $\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$

There is a result of the form: a function $u \in H^1(0,T;L^2)$ if and only if $$\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$$ holds for all $h \in [0,T]$. I have only seen one place ...
0
votes
0answers
86 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
2
votes
0answers
46 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
0
votes
1answer
35 views

Some extensions of Weyl's lemma

The Weyl's lemma said that: If $u$ is a continuous function on the open set $\Omega$ such that it satisfies $\Delta u=0$ in the distributive sense, that is $$\int_\Omega u \Delta\phi = 0$$ for all ...
1
vote
1answer
33 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
0
votes
0answers
41 views

Reference request on Linear partial differential equations

I'm looking for recent references on linear partial differential equations, containing topics similar to L. Nirenberg's Lectures on Linear Partial differential equations, namely discussion on ...
0
votes
1answer
36 views

weak solution of Dirichlet problem in Lipschtiz domain with non zero boundary data

Let $\Omega$ a bounded and open with Lipschitz boundary. I know that exists the trace operator in the case of this $\Omega$. My question is : When $\Omega$ is bounded and open with Lipschitz ...
1
vote
0answers
21 views

Symmetrisation of system of conservation laws

Suppose we have a system $$\partial_t \rho +\nabla_x\cdot \upsilon(\rho)=0,$$ where $x\in \Bbb R^3$, $\rho:(t,x)\to \Bbb R^5$, $\upsilon: \Bbb R^5\to \Bbb R^3$. Suppose also that there exists a ...
5
votes
2answers
109 views

Solving wave equation

I want to solve the following PDE: $$\begin{align} u_{tt}&=c^2u_{xx}-\gamma u_x, \quad 0<x<1, \quad t>0,\\ \\ u(0,t)&=u(1,t)=0, \\ u(x,t=0)&=x(1-x),\\ u_t(x,t=0)&=0. ...
3
votes
3answers
46 views

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 ...
2
votes
1answer
97 views

$H^{-1}(\Omega)$ 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 ...
0
votes
1answer
40 views

Textbook for partial differential equations with the tools of complex analysis

Is there a textbook in partial differential equations that has a lot of solved excercises by using complex analysis tools?