# Tag Info

• 463
1 vote

### Maximum principle of strong solution of linear parabolic equation in $\mathbb{R}\times [0,T]$

I think the integration by part of the second order derivative should be $$\int^R_{-R}\partial_x^2 (u-u^*)(u-u^*)^+dx= [\partial_x^2 (u-u^*)(u-u^*)^+]^R_{-R} -\int^R_{-R}|\partial_x(u-u^*)^+|^2dx.$$ I ...
• 437

### Uniqueness and continuous dependence on the data of Heat equation.

To address the lack of connectedness when applying the strong maximum principle: Since $U\subset \mathbb R^m$ is open and $\mathbb R^m$ is connected, it follows that $U$ is locally connected from ...
• 12.5k
Accepted

• 30.7k

### Is $W^{k,p}(\mathbb{R}^n)$ equal to $W^{k,p}_{loc}(\mathbb{R}^n)$?

These spaces $W^{k,p}_{loc}(\Omega)$ and $W^{k,p}(\Omega)$ are never equal for open sets $\Omega$. The problem is that for these loc-spaces the behavior close to the boundary (and at infinity) cannot ...
• 49.2k
1 vote

### Example of an integrable function $f$ such that $|\int(1+x^2)f(x)\,dx|<|\int f(x)\,dx|$

Let $f(x) = 1_{[0, 1)}(x) - 1_{[1, \frac{3}{2}]}(x)$. It is clearly in $L^1$. The LHS equals $\frac{5}{12}$ while the RHS equals $\frac{1}{2}$.
• 5,120

### d'Alembert's solution to the wave equation via Fourier Transforms

\begin{align} \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\hat g(k)}{k}\sin (kt) e^{ikx}dk &=\frac{1}{4\pi i}\int_{-\infty}^{\infty}\frac{\hat g(k)}{k}\left[e^{ik(x+t)}-e^{ik(x-t)}\right]dk \\ &...
• 9,447

• 9,447
1 vote

• 1,815
Accepted

### Piecewise Continuity enough to ensure finite Fourier coefficients?

The function $s(x)=\dfrac{1}{x^2}$ is not piecewise continuous on $[-\pi,\pi]$, because it is not even defined at $x=0$. Even if one defined $s(0)=c$ for some $c\in\mathbb{R}$, this function is still ...
• 4,931

### Fundamental solution of heat equation in a ball with homogeneous Dirichlet boundary condition

For the standard heat equation ($\partial_t u-\Delta u=0$), there is an explicit solution formulated in Gegenbauer polynomial when dimension is not less than 3. See Theorem 2 in Hsu, P. (1986). ...
• 46

### On exact differential equation

You are looking for the Poincare lemma and you also need the assumption that you are trying to solve that exact DE on a star-shaped (convex or simply connected will also do the trick). Let us assume ...
• 4,360

• 12.5k
1 vote
Accepted

• 5,492

### Does the inequality $\iint_{\mathbb R^2} \frac{|f(x)|^2}{|x-y|^a} \ dy\,dx\le k ||f||_{L^2(\mathbb R)}^2$ hold?

Your estimate is a special case of the weak Young-inequality (used in Potential theory) which follows from the Hardy-Littlewood-Sobolev inequality, see the first answer of https://mathoverflow.net/...

• 19.3k

• 2,276

### Proving $\int\limits_{Q} u^2 dx ≤ C(n) \int\limits_{Q} (u^2_{x1} + · · · + u^2_{xn}) dx$ over a cube $Q$

One possible proof would be via developing $u$ into its Fourier series and then using simple formula for second norm (so called Parselval's theorem). In case of $n=1$, say $u = \sum a_ke^{2{\pi}kx}$, ...
• 117

### What can semigroup theory do better in the study of PDEs compared to alternative methods?

I have not yet understood what does the theory offer for the study of PDEs, that other methods do not provide? A very important aspect is study of the asymptotic behavior by estimating the resolvent ...
• 18.9k
1 vote

### Implementing Boundary Conditions at Infinity in Numerical Method

The way to approach this numerically is to implement a perfectly matched layer boundary condition. When you are doing a graphical analysis of your numerical solution, you can only do it in a finite ...
1 vote

### weakest condition on $\Omega$ such that $W^{1,p}_0(\Omega)= \lbrace u: \tilde{u} \in W^{1,p}(\mathbb{R}^N)\rbrace$

This is too long for a comment. Initially, I thought that this holds for all open (and bounded) sets. However, the following is a counterexample. Let $\Omega = (-1,1)^2 \setminus \{0\}\times (0,1)$. ...
• 31.4k
1 vote
Accepted

### Solving the Poisson equation $-\Delta u=f$ on a domain $G$

Yes, you are completely right. However, I think there is an issue that $f \in C_c^1(\mathbb{R}^n)$ and not $f \in C_c^2(\mathbb{R}^n)$. If the latter was the case, you could look up all the details in ...
• 4,360

### semi group and perturbation trick

I guess you will need some regularly for $A$ (for $\mathrm{div}$?) to ensure the well-posedness. The classical strategy for the proofs is as follows: Existence: you prove that the governing operator ...
• 2,248
Accepted

### Existence of an extension operator $E: W_0^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$

This is a very nice question. It appears to me that the answer is in general, no. Moreover, it appears to be an open problem determining geometric conditions on $\Omega$ for which such an extension ...
• 7,034

### Non-linear first order PDE

Here is an alternative solution, using the method of characteristics. Let's rewrite the PDE as $$F:=xp^2+q-y=0 \qquad(p=u_x,\, q=u_y). \tag{1}$$ Solving the Lagrange-Charpit equations, we obtain \...
• 9,447

### How to find the characteristic form of coupled PDEs?

Just follow example 1, p. 117. Let $a^{2} = A$. Then the linear combination \begin{align} l_{1} (k_{t} + \omega_{0}' k_{x} + \omega_{2} A_{x}) + l_{2} (A_{t} + \omega_{0}' A_{x} + \omega_{0}'' A k_{x})...
• 4,249

### $[a,b]$ has smooth boundary???

Yes, it is smooth. To see this, you can use one of the classical characterizations of smooth sub-manifolds of $\mathbb R^n$. Then, the set $\{a,b\}$ is a smooth ($C^\infty$) manifold of dimension $0$.
• 2,248
Accepted

### How can I find out if this 1st order quasi-linear pde has *no solution*

The answer from Toby Saunders-A'Court begins with an assumption. The assumption is found valid at the end. So his answer is correct and this is by far the simplest way to solve the problem. More ...
• 66.3k
1 vote
Accepted

### Calculating the Divergence of a Function on Unit Ball

The divergence, denoted here as $\operatorname{div}$, is the divergence on $\mathbb S^{n-1}$. Let $M^{n-1}$ be an oriented hypersurface in $\mathbb R^n$ with unit normal $\nu$ and $F:M \to \mathbb R^n$...
• 7,034