# Tagged Questions

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

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### Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: $$\operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu,$$ where $\mu$ is ...
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I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE): $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(... 0answers 25 views ### Wave equation in a cube Is it possible find a computable solution to the following homogeneous wave equation problem: Let \mathcal{C}=\{(x,y,z)\in \mathbb{R}^3, 0< x,y,z < 1\}  be the open unit cube. Find u such ... 0answers 21 views ### Poincaré's inequality proof for u \in W_0^{1,p}(\Omega). I am trying to prove Poincare's inequality for u \in W_0^{1,p}(\Omega), where \Omega \subset \mathbb{R}^n is an open bounded set and 1 \leq p < \infty. This is Poincare's inequality: ||... 0answers 35 views ### The 3rd term of the energy estimates in chapter 7 Evans PDE Hi I wonder someone could help me check my understanding of getting an inequality of the estimate for the 3rd term \|u'_m\|_{L^2(0,T;H^{-1}(U))} correctly. This inequality need to be checked is ... 0answers 42 views ### Fractional powers of Markov generators Let H be the generator of a symmetric Markov semigroup on L^2(\mathbb{R}^n). Why the fractional power H^\alpha (defined on a proper domain) with 0 < \alpha < 1 turn out to be the ... 1answer 26 views ### How to find other equations of lagrange for the Initial Value Problem Find the solution of the Initial Value Problem (x-y)\dfrac{\partial u}{\partial x}+(y-x-u)\dfrac{\partial u}{\partial y}=u where u(x,0)=1 . My try: \dfrac{dx}{x-y}=\dfrac{dy}{y-x-u}=\dfrac{... 0answers 29 views ### PDE with Stochastic Coefficients Does anyone have reference suggestions for pde's with stochastic coefficients? I've seen many papers on more advanced problems, but it would be great to have a reference discussing the basic theory ... 0answers 29 views ### A comparison principle for a nonlinear parabolic PDE We know the following comparison principle holds for the diffusion equation: Suppose that u(x,t) and v(x,t) satisfy \begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \... 1answer 40 views ### Strange PDE solution Given the linear equation$$u_t -xt u_x = x$$x\in\mathbb{R}, t>0, with IVP u(x,0)=u_0(x), my solution comes to u(x,t) = u_0(xe^{t^2/2})+xt, but Maple gives a much more complicated solution ... 0answers 39 views ### Eigenvectors of matrix equation AX+XB^\text{T}=\lambda (CX+XD^\text{T}) We have the matrix eigenvalue problem$$AX+XB^\text{T}=\lambda (CX+XD^\text{T})$$Where \lambda is the eigenvalue, X is m\times n and plays the role of the eigenvector, and A, B, C, and ... 0answers 49 views ### Hadamard counterexample Dirichlet-problem Good afternoon, I try to understand the follwoing counterexample of Hadamard: The classical solution of  \begin{cases} \Delta u = 0 ~~\text{ in }\Omega\\ u=g \text{ auf }~~ \partial \Omega \end{... 0answers 22 views ### PDE realated to heat equation with exponential additive term I want to solve a PDE realated to heat equation with exponential additive term$${\partial u\over\partial t}={1\over x^2}{\partial\over\partial x}(x^2{\partial u\over\partial x})+e^u$$I dervived ... 1answer 38 views ### Confusion with changing variables in second order DE So in my physics assignment, we're given Schrodinger's equation:$$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2}+e\xi x\psi=E\psi$$We're asked to substitute a function of the form w(x)=Ax+B to arrive at ... 1answer 144 views ### Solving this non-linear PDE (which reminds of a linear parabolic PDE) Problem: consider the following PDE:$$-u_t=\mbox{sign}(u) u_x+ \frac{1}{2}u_{xx},$$with some boundary condition u(T,x)=\delta_a(x)-\delta_{-a}(x), a>0 fixed, being \mbox{sign}(u)\in \{-1,1\}... 0answers 38 views ### Dirichlet Problem in Stochastic PDE Section of Probability Textbook. I recently started learning about stochastic calculus and stochastic PDEs, and I don't know where to begin with some of the problems in my textbook. Any help with the following or a push in the right ... 0answers 51 views ### On spectrum of periodic boundary value problem Consider the following boundary value problem on the infinite strip (-\infty,\infty)\times[0,1] w/periodic conductivity \gamma(x,y)=\gamma(x+2\pi,y)>0:$$\begin{cases} \operatorname{div}(\gamma\...
How did the author find the test function $\varphi$? Thanks