Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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57 views

What is the difference between Deformation technique and Ekeland's variational principle?

What is the difference between Deformation technique and Ekeland's variational principle to approach Mountain Pass theorem ?
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100 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
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45 views

Attach term to solution of PDE(perturbation theory)

Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
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76 views

A “straightforward” inequality.

In section 5.5 page page 117 of Fanghua, Quing's book, in order to obtain $W^{2,p}$ estimates the idea is to show that for $p \in (1,\infty]$ the condition $\theta \in L^p(\Omega)$ implies $D^2u \in ...
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139 views

How to solve the advection equation with spiral motion

The advection equation is : $$\frac{\partial f(x,y,t)}{\partial t} + \nabla_{(x,y)} \cdot (A f)= 0$$ With initial condition $f(x,y,0) = f_0(x,y)$. If the vector $A$ is constant, ie. $A = ...
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108 views

Green's Function Divergence

Given a domain $ \Omega \in \mathbb{R}^2 $, and a PDE of the form $ L = a(x) \partial_x^2 + b(x) \partial_y ^2 $ for $ x \in \Omega $ , the green's function $ G(x,y) : \Omega \times \Omega \rightarrow ...
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79 views

Adjoints of an operator

In a paper I'm reading the discussion centres on operators of the form \begin{equation} \mathbf{B} = (-1)^{m+1} \Delta^m_y + \frac{1}{2m}y\cdot\nabla_y \end{equation} Apparently this is symmetric ...
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87 views

Integration by Parts for PDEs

I'm reading a paper on PDEs in preparation for some research. In it, integrals like this appear repreatedly: $$ I(x,t) = \int_{|y|>1} e^{-i\alpha xy} \frac{e^{i\beta y^2}}{(1+y^2)^m} dy. $$ Here ...
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45 views

Classical Parabolic theory (PDE)

I am reading an article and I almost done but I don't understand an argument on page 8: It is known from the classical parabolic theory that in the case when the nonlinearity grows no faster than ...
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91 views

Is this Stokes problem well-posed?

I am solving Stokes problem: $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ in a domain that's bounded by two surfaces - a cuboid and a small ...
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150 views

Checking initial condition of PDE is satisfied in Galerkin method

The following is an argument I read, it is part of a proof of checking that the solution given by the Galerkin method satisfies the initial conditions. From our PDE $$\langle u', v \rangle_{V',V} + ...
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46 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
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75 views

Confused by a (standard?) gradient estimate.

This sort of inequality appears frequently in a paper I'm reading, but never with any justification. Take $u(x) \in H^{1}(B(x_{0}, r))$ and a cutoff $\phi(x) \in C_{0}^{\infty}(B(x_{0}, r))$ where ...
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173 views

Reference: Hölder regularity for linear elliptic equation, Neumann problem

I checked the Book of Gilbarg Trudinger on elliptic equations for (Hölder)-regularity results on weak solutions for elliptic pde. In particular, my equation is of the form $$\nabla \cdot ...
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117 views

Prove the equivalence of the solution existence of Stokes equation's variational problem to the original problem

I have this exercise. We put $X = L^2(\Omega)^4 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$ $$a : X \times X \rightarrow \mathbb{R}; ((\sigma,p),(\tau,\alpha)) \rightarrow (\sigma \otimes \tau)$$ ...
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266 views

Weak solution to PDE boundary value problems

Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset ...
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53 views

Timestepping PDE with positive eigenvalues

I'm trying to numerically solve a PDE, namely: $$ \partial_t \binom{u(x, t)}{v(x, t)} = x \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right) \cdot \partial_x \binom{u(x, t)}{v(x, t)} ...
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73 views

Why can I write the curve shortening flow system as..

Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
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58 views

Regularization of solutions from a quasilinear equation.

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Let $-\Delta_p :W_0^{1,p}\to W^{-1,p'}$ be the $p$-Laplace operator, i.e. $$\langle-\Delta_pu,v\rangle=\int_\Omega ...
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195 views

Regularity of weighted p-Laplace equation up to boundary

Take the weighted p-Laplace equation, $\nabla \cdot (\gamma |\nabla u|^{p-2}\nabla u) = 0$, with smooth Dirichlet boundary values on some smooth and bounded domain. Suppose the weight $\gamma$ is also ...
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504 views

About finding the general solution of first-order totally nonlinear PDE with two independent variables

Recently I feel very interested about finding the general solution of first-order totally nonlinear PDE with two independent variables. However, most PDE books only discussed little about finding the ...
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125 views

Is this pde Elliptic?

I have seen that there is a classification of elliptic pde's. It says the pde $$au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y+f =0$$ is elliptic if $b^2 - ac < 0$. There is another definition that is ...
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157 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
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120 views

Boundedness for Reaction Diffusion BVP with Arbitrary Exponent $\alpha$

Let $U\subset\mathbb{R}^{n}$ be open, $U_{T}$ and $\Gamma_{T}$ be the parabolic cylinder and boundary of $U$ for arbitrary $0\leq t\leq T$, respectively, and suppose $u$ solves $$ ...
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219 views

Solving systems PDE's

I have a bit of trouble solving a system of first order PDE's, that I get by solving a boundary issue problem in gravitation (here). I have six equations: ...
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129 views

invertible operator Sobolev space

Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
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172 views

Why is this function space dense in this Bochner-type space? (Rogers and Renardy)

Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$ Renardy and Rogers says that 1) Functions of the above form are dense in ...
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Is a solution to some partially differential equation homeomorphic (or diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
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110 views

an application of implicit function theorem?

I need to proof the following, could any one help me to proof step by step? $(t,\epsilon)\mapsto F(t,\epsilon):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a twice continuously differentiable ...
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218 views

Show that the convolution of a spherically symmetric function with the heat kernel is also spherically symmetric

Let $\phi \colon \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Furthermore, suppose that $\phi$ is spherically symmetric (i.e., assume $\phi(Tx) = \phi(x)$ for every orthogonal ...
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293 views

Wave Equations Concept questions

My Engineering Mathematics teacher have a very novel method of teaching. He thinks that while it's all good and well to learn the analytical side of math, he stresses more on the theory, than the ...
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80 views

Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators". The motivating problem for this is to find an approximate kernel ...
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268 views

When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
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245 views

how to uncouple and unreduce a system of first order PDEs

Suppose we are given a system of first order PDEs with constant coefficients. In particular, suppose we are given $k$ PDEs for $u_1,u_2, \dots u_n$ with respect to independent variables $x_1,x_2, ...
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127 views

2nd Order Elliptic PDEs with functional BCs

I'm interested in studying linear second order elliptic PDEs with boundary conditions that are functionals of the solution and possibly its derivative. For example, \begin{align} \nabla^2 u(\vec{x}) ...
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110 views

An existence Theorem

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
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429 views

Drum membrane wave equation general solution (non-symmetrical)

I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, ...
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498 views

Characterization of Cauchy-Riemann operator

Let $U \subset \mathbf C$ be an open subset of the complex plane and suppose we have a differential operator of order 1, $L: \mathcal C^{\infty}(U) \to C^{\infty}(U)$ such that $Lu = 0$ if and only ...
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1k views

What is the difference between the domain of influence and the domain of dependence?

When analysing the wave equation $$u_{tt} = c^2 u_{xx}$$ in my PDE's module, I understand the 'domain of dependence' which is where the value $u(x_0,t_0)$ is only depends on the initial value of $x$ ...
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216 views

Elliptic PDE regularity

I have $u \in H^1_0(\Omega)$ where $\Omega$ is bounded which solves some elliptic PDE of the form: $$-\Delta u + h(u) = f$$ in $\Omega$, where $f \in L^2(\Omega)$ $$u = 0$$ on $\partial\Omega$, say. ...
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164 views

Trying to solve Navier equations with Maple

I've been trying for ages now to get this to work, it's a system of partial differential equations (Engineering, Plate theory) $$pde1: = \frac{{{\partial ^2}}}{{\partial {x^2}}}u\left( {x,y} \right) ...
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155 views

Integral representation of directional derivative

We consider the Dirichlet problem $$ \begin{array}{c} \Delta\psi(x) + v(x)\psi(x) = 0, \;\;\; x \in D \\ \psi|_{\partial{D}} = f \end{array} $$ in some bounded region $D$ with smooth ...
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170 views

Using games to approximate solutions to PDE's

Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "Partial ...
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265 views

The canonical form of a nonlinear second order PDE

Can anyone help me find the canonical form of $$x^2u_{xy} - yu_{yy} + u_x - 4u = 0?$$ I don't know how to solve it because $a = 0$. I just got that it's hyperbolic since $a=0$ , $b =(x^2)/2$, $c= ...
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110 views

Solving a two-dimensional system of conservation laws

I have $$\begin{align} u_x + u_t &= 0\\(\rho u)_x + \rho_t &= 0 \\ \left(Eu\right)_{x}+E_{t}+pu_{x}&=0\end{align} $$ satisfying these boundary conditions: $u\left(x,0\right)=x,\ ...
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30 views

Mixed Dirichlet-Neumann eigenvalue problem

Let $\Omega\subset\Bbb R^2$ be a bounded $C^2$ domain. Let $\partial\Omega=\partial\Omega_1\cup\partial\Omega_2$. Does anyone know about the existence of eigenvalues and eigenfunctions for the ...
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39 views

$u=xf(xy)$, show that $xu_{xx}-yu_{xy} = 0$

I need to show that: $$xu_{xx}-yu_{xy} = 0$$ when $$u=xf(xy)$$ So, I did: $$u_x = xyf_x(xy)+f(xy) \implies $$ $$u_{xx} = xy^2f_{xx}(xy)+2yf_x(xy)$$ and $$u_{xy} = ...
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20 views

$1$st Order PDE

I was solving 1st order PDE which was \begin{equation} (x^2-y^2-z^2)p+2xyq=2xz. \tag{1} \end{equation} I had tried to solve this. Please tell me whether it is correct or not. ...
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17 views

How does one find the coefficients in the solution to the Laplace equation?

I'm reading this. Equation (522) gives the general solution to the Laplace equation. What I'm stuck about, is how to determine the coefficients $a_m$, $\beta_m$, and $\theta_m$ for non-trivial ...
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23 views

Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...