Questions tagged [partial-differential-equations]
Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.
24,346 questions
0 votes
0 answers
40 views
How should I find the general solution of the partial differential equation $u_{xx}+4u_{yy}=0$ and what's the correct answer to this given PDE?
Find the general solution of the partial differential equation $u_{xx}+4u_{yy}=0$. Here's my work: When it comes to finding the general solutions of this type of partial differential equations, does ...
- 59
0 votes
0 answers
30 views
Spectral decomposition of the Neumann-Laplacian
I'm currently trying to find a spectral decomposition for the Laplace Operator such that the eigenfunctions are a part of $H^2_N$ := $\{u \in H^2 \,|\,\frac{\partial u}{\partial \nu} = 0 \,\,\text{on} ...
- 1
1 vote
1 answer
86 views
The Gateaux derivative and the Jacobian
Question Is it accurate to claim that the Gateaux derivative is the continuous linear operator represented by the Jacobian matrix? That is, the Jacobian is the finite dimensional equivalent of the ...
- 267
1 vote
0 answers
23 views
Composition in Sobolev Spaces [duplicate]
Let $\Omega \subset \mathbb{R}^n$ a bounded domain, $f:\mathbb{R} \to \mathbb{R}$ of class $C^1$ such that $f(0)=0$. Let $u \in L^{\infty}(\Omega) \cap W_0^{1,p}(\Omega)$ for some $1 \leq p <+\...
- 255
2 votes
0 answers
45 views
Parabolic Partial Differential Equations
Firsts things first, a bit of context: I'm undergraduate student who, still, haven't take a proper class on PDEs. I'm fine with basic concepts, but I'm completely foreigner to the methods of solivng ...
- 1,182
1 vote
0 answers
35 views
Existence and uniqueness of a solution to a Ricci tensor-based boundary value problem [closed]
Let $g$ be a Riemannian metric on a domain $\Omega$ with boundary $\sigma$, and the Ricci curvature tensor is given by $$ R = [\, \partial \Gamma \,] + [\, \Gamma \Gamma \,], $$ where $$ {[\, \partial ...
0 votes
0 answers
55 views
How to show $\int_{\mathbb R^n} |Du|^2 dx = \int_{\mathbb R^n} |y^2| |\hat u|^2 dy$?
Pictures below is from Evans' PDE, I want to calculate the red line. $\hat u$ is the Fourier transform of $u$, namely $$ \hat u(y) = \frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n} e^{-i x\cdot y} u(x) dx . $...
- 6,964
1 vote
1 answer
87 views
Fractional Sobolev space $H^\frac{1}{2}$ is equal to the interpolation space $(L^2,H^1)_{\frac{1}{2},2;K}$
I want to show that the fractional Sobolev space $H^\frac{1}{2}$ is equal to the interpolation space $(L^2,H^1)_{\frac{1}{2},2;K}$. In particular I would like to use the K-method as described in the ...
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