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

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About Semiclassical Analysis and other

I read something about this theory. I honestly do not care to find out the link between quantum mechanics and general relativity, because it's too much for me. But I have seen that there are still ...
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21 views

Fundamental Solution Cauchy-Riemann Operator

Consider the linear partial differential operator $L=\partial_{x}+i\partial_{y}$ acting on distributions on $\mathcal{D}'(\mathbb{R}^{2})$. Folland writes that a fundamental solution of this operator ...
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Show that $||u||_{L^{1}(I)} \le \frac{1}{k} ||f||_{L^{1}(I)}$

Suppose there exists a unique $u \in H_0^{1}(I)$ satisfying $$\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$$ Show that $$||u||_{L^{1}(I)} \le \frac{1}{k} ||f||_{L^{1}(I)}$$ In the ...
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1answer
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prove that there exists a unique $u \in H_0^{1}(I)$ satisfying $\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$

Let $I=(0,1)$ and fix a constant $k \gt 0$. Given $f \in L^{1}(I)$ prove that there exists a unique $u \in H_0^{1}(I)$ satisfying $$\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$$ For ...
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1answer
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Two definitions of set of class $C^1$, given in books by Brezis and Evans

I am reading the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. It is given the following definition of an open set of class $C^1$, which I find it hard ...
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6 views

Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
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9 views

Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
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2answers
21 views

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with ...
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A nice question related to method of characteristics

Let $ \alpha$ be real number and $h=h(x)$ be a continuous function in $\mathbb{R}$.Consider following initial value problem: $$yu_x + xu_y=\alpha u, u(x, 0) =h(x) $$ Then a) Find all points on ...
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17 views

Complete integral solution of first order PDE [on hold]

Find a complete integral of $4uu_{x}-u_{y}^{3}=0$, and then that solution which satisfies $u=4at$ on $x=0$, $y=t$
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1answer
54 views

What is precisely the definition of Elliptic Partial Differential Equation?

I am reading about the Variational Method to solve Elliptic PDEs but I have been unable to find a precise definition of the concept Elliptic PDE. Of course, I am aware of the definition given in ...
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1answer
18 views

On the trace theory and restrictions of Sobolev space functions

Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...
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14 views

Transport PDEs with mixed linear and nonlinear terms and distribution solutions

This question is concerned with the theory of solutions of first order transport-type PDEs that are linear in some variable and nonlinear in others. E.g. this beauty: $\frac{\partial u}{\partial ...
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13 views

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} ...
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Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class ...
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1answer
29 views

Find the general solution of the pde. [on hold]

Find the general solution of the equation for $v=v(x,y)$: $$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}=2xy(x^2-y^2)$$
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1answer
19 views

Domain of dependence and energy in the wave equation

Where does the second summand come from in the derivative expression? I am fairly certain it has to do with the fact that the boundary is dependent on $t$, but can someone provide either intuitive ...
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20 views

Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...
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5 views

A version of Hörmander multiplier theorem

Let $m>n/2$ be an integer. Let $h\in H^m_{loc}(\mathbb{R}^n)$ satisfy that $\displaystyle \exists M>0,\forall R>0,\sum_{|\alpha|\le m}\int_{\frac R2\le|w|\le2R}R^{2|\alpha|}|\partial^\alpha ...
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What do they mean by X1 and X2 satisfy same Robin boundary condition at $a$ and at $b$? [on hold]

What do they mean by X1 and X2 satisfy same Robin boundary condition at $a$ and at $b$? Does it mean $X_1'(a)+pX_1(a)=X_2'(a)+pX_2(a)$ and $X_1'(b) +pX_1(b)=X_2'(b) +pX_2(b)$ for some p?
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1answer
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Intuition behind boundary conditions for PDE

Suppose that I am trying to model the spread of heat through a $1$ dimensional rod of length $L$ in meters with $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}$$ where $u(x,t)$ ...
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24 views

Find the general solution of this pde?

I have a final in my PDE Class and needs some help with a review problem. Find the general solution of the equation for $v=v(x,y)$: $$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial ...
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1answer
23 views

Meaning of generators in Lie Algebra of PDE

Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators, $X_1=\partial_x,\quad ...
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1answer
21 views

Fourier decomposition of solutions of the wave equation with respect to the spatial variable

Say I have a wave equation of the form $$\nabla^{2}f(t,\mathbf{x})=\frac{1}{v^{2}}\frac{\partial^{2}f(t,\mathbf{x})}{\partial t^{2}}$$ which is clearly a partial differential equation (PDE) in ...
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1answer
24 views

Equivalent ways to study perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates a $C_0$ ...
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22 views

Nonlinear cauchy problem for PDE [on hold]

Let $u=u(x,t)$ be the solution of the cauchy problem $\frac{\partial u}{\partial t}+(\frac{\partial u}{\partial x})^{2} =1 , x\in R, t>0 \\ u(x,0)=-x^2, x\in R \\$ Then how to find the solution ...
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heat equation-calculate the temperature of an bar

let an bar of lenght 50 cm, and temperature on t=0 is 100 degree. The question is calculate the degree on the middle of the bar. So i try to write the heat equation: $\dfrac{\partial u}{\partial t}= ...
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1answer
28 views

Solving PDE $v_t(t,x) + \frac{1}{2} v_{xx} (t,x) = 0$

We are given a PDE with $$v_t(t,x) + \frac{1}{2} v_{xx} (t,x) = 0$$ $$ v(T,x) = x^2 $$ for $0 < t \leq T$ and $x \in \mathbb{R}$ So far I have found that using the Feynmann-Kac equation, we get ...
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Computing a Green's function - where did I go wrong?

This is from a homework problem that was recently returned to me in a numerical analysis course. The grader even noted that he didn't know where I went wrong but the solution was marked as incorrect. ...
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Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
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How to prove the sufficient condition for a PDE has positive eigenvalues

Strauss' textbook says it is $\left[f(x)f'(x)\vphantom{\dfrac11}\right]_{x=a} ^{x=b}<0$, or $=0$. the assumption is $f''(x)=-\lambda f(x)$ for real $\lambda$. they told me to use Green's first ...
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1answer
38 views

Singular Perturbation Theory problem for slightly unstable pde

I'm learning about singular perturbation theory for solving an approximate solution to a pde and I'm a little confused on how to apply it. The problem I'm trying to work on is $\frac{\partial ...
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44 views

Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$

In the plane find two solutions of the initial-value problem $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$. I think we get to use the method of characteristics But I am not ...
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1answer
21 views

Integrals with functions as bounds

How to calculate integral such as $$\int_{g(χ)}^{φ(χ)} f(s) \, ds$$ where $F'(s)=f(s)$ Integrals like this appear often in PDE's .I'd like to know the whole theory i mean if there is a formula how ...
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When is a symmetric hyperbolic system a second order scalar equation?

Given a linear symmetric operator of the form $$A^\mu \partial_{\mu}v+Bv=F$$ where $v\in C^{\infty}(\mathbb{R}^m ,\mathbb{R}^n)$ is a solution of the system and $A^\mu, B$ are $n \times n$ matrices ...
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PDE, radially symmetric function

I got to show that the minimal surface $$(1): \nabla (\frac{\nabla u(x,y)}{\sqrt{1+|\nabla u(x,y)|^2}})=0$$ for radially symmetric functions can be wrtten as $$(2): \partial ...
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Solution to Laplaces

I am trying to show that the following is a solution to Laplaces equation. $v(x) = c_2*Ln(|x|), x \in R^2$ where $c_2$ is an arbitrary constant. And $x \neq 0$ . Im just stuck on how to get started ...
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Variable seperable method [on hold]

Find a solution of the equation $$\frac{∂^2u}{∂x^2}=\frac{∂u}{∂x}+2u$$ in the form $u=f(x)g(y)$. Solve the equation subject to the conditions $u=0$ and $∂u/∂x = 1+ e^{-3y}$ when $x=0$ and for all ...
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Question about asympotic behavior of $\frac{1}{s}\int_0^s u(x,t) dt$ .

I am just reading a paper, in the final theorem, the author wants to prove that $u(x,t)$ converges to some $v(x)$ in the $L^2$ norm as $t$ $\to$ $\infty$. But in the proof, he defines a ...
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Seeking help with finding the general solution of this differential equation

I am trying to find the general solution of the following equation. $$\int_0^\infty \frac{\partial f(x, t)} {\partial t} \sin(x \xi) \, dx = \xi \int_0^\infty f(x, t) \cos(x \xi) \, dx -\alpha \xi ^2 ...
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Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
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1answer
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Solve transport equations by using Laplace transform

I'm trying to solve rather formally one-dimensional transport equation: $$ u_{t}+cu_{x}=0\quad\text{in $(0,\infty)\times(-\infty,\infty)$} $$ with an initial data $u_{0}$, which is bounded and ...
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1answer
19 views

Estimation of gradients in Poisson's equation

I am trying to show the following result. Let $D\subset\Bbb R^3$ be a bounded open set with smooth boundary. For any $f\in H^{-1}(D)$, let $\phi$ be the unique weak solution to the following ...
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Diffusion/Heat equation, weak maximum principle

Consider the problem: $$u_t -div(A(x) \nabla u) +a(x) u = f $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) $$ and its variational formulation $$ \lt \dot{u(t)},v \gt_* + B(u(t),v;t) =(f,v)_{L^2} \quad \forall ...
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1answer
15 views

Does uniformly boundness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this ...
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1answer
40 views

Kato's inequality

Let u be a smooth function defined in a Riemannian manifold $(M,g)$. The well known Kato's inequality states $$|∇|∇u||^2≤|∇^2u|^2$$ where $∇^2$ represents the Hessian operator of $M$. I would like ask ...
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Solving $4u_{tt}-3u_{xt}-u_{xx}=0$

Solving $\begin{cases} 4u_{tt}-3u_{xt}-u_{xx}=0\tag1\\u(x,0)=x^2\quad\text{and}\quad u_t(x,0)=e^x\end{cases}$ in $\mathbf R\times\mathbf R_{>0}$ First I factorized and get for the first line; ...
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8 views

unicity solution of partial differential equation

let the following problem: $$ \begin{cases} \dfrac{\partial^2 u}{\partial t^2}= a^2 \dfrac{\partial^2 u}{\partial x^2}\\ u(0,t)=u(l,t)=0\\ u(x,0)=f(x)\\ \dfrac{\partial u}{\partial x}(x,0)=g(x) ...
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15 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...