Tagged Questions

-1
votes
1answer
51 views

How to prove the following equality in $\,L^2\,$?

A book has the following: Because the funcions $\;e^{ikt},\;\;k=-n,-(n-1),\cdots,0,1,2,\cdots,n\;$ are orthogonal, we have ...
3
votes
1answer
45 views

Elliptic regularity - nonlinear case

Let's say I have a weak solution $u \in H^1(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, to the equation $$ \Delta u = e^u, $$ let's also assume that $e^u \in L^\infty(\Omega)$. Does it follow that ...
1
vote
0answers
51 views

Physical interpretation of a wave equation

Do exist any physical interpretation of (several) boundary conditions like : $$y=g_1\ne 0,\; \Delta y=g_2,\; ..=\Delta^{k-1} y =g_k,\; \mbox{on} \; \partial \Omega,\; k\ge 2$$ for the wave equation ...
2
votes
0answers
31 views

Riemann mapping between arbitrary triangles

Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)? Comment---I look for the conformal equivalence of interiors promised ...
7
votes
2answers
104 views

Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
2
votes
1answer
44 views

Fourier Transform on Infinite Strip Poisson Equation

Im trying to solve the following Poisson equation: $$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$ $$u(x,0) = 0,\ u(x,1) = 0$$ $$u(x,y) \to 0\ ...
0
votes
1answer
44 views

relation between Holder continuous and weakly differentiable for the coefficients of a pde

I am reading a book on pdes and the author gives the definition of the weak solution using the adjoint operator. For that expression to make sense, for the case of second order elliptic equation one ...
0
votes
1answer
54 views

Differentiable but not Absolutely continuous

Please give an example (if it exists) for a function which is differentiable everywhere but not absolutely continuous.
2
votes
0answers
54 views

Construction of Monotone function which is differentiable on the given set

Given a set $A \subset \mathbb{R}$ of measure 0, is it possible to construct a monotone function whose set of non differentiable points is $A$ ?
2
votes
0answers
46 views

Sobolev Spaces: The difference between $W^{k,p}$ and $W^{k,p}_0$

Let $U$ be an open set in $R^d$. I am confused about the differences between $$W^{k,p}(U):=\{u\in L^p(U): D^{\alpha}u\in L^p(U) \text{ for all } |\alpha|\le k\}$$ and ...
0
votes
2answers
45 views

Boundedness of a Solution Operator

Let $v \in L^2(\partial \Omega)$ and define $S(v) := y$ where $y$ satisfies $-\Delta y + y = 0 $ in $\Omega$ and $\frac{\partial y}{\partial \nu} = v$ on $\partial \Omega$. I need to show that $S$ is ...
1
vote
1answer
65 views

Alternative complete bases for Fourier Series.

Knowing that $$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$ are complete systems in $L^2(0,\pi)$. How ...
1
vote
0answers
37 views

exercise : capacity theory

Can someone give me a hint to verify this affirmation? "Any set $E \subset R^n$ of $(p, \theta) - $ capacity zero is contained in a Borel set $\tilde{E}$ of $(p, \theta) - $ capacity zero; in fact, ...
0
votes
1answer
39 views

theorem in capacity theory

I am trying to understand the proof of a theorem of capacity theory the theorem can be found in the book "nonlinear potential theory of degenerate elliptic equations" the authors are Juha Heinonen, ...
1
vote
0answers
25 views

Weak Derivative is $0$ [duplicate]

If the weak derivative of some function $u$ on a open connected set is 0. Does it mean that $u$ is constant almost everywhere on the given set?
1
vote
0answers
93 views

Viscous Burgers Equation existence of solution

Can some one prove the local existence of solution for Viscous Burgers equation using fixed point techniques. The equation is as follows : $$ \frac{\partial u}{\partial t} - \bigtriangleup u + ...
3
votes
1answer
65 views

Confused about Proof of Thm 4.9 Gilbarg Trudinger

Thm 4.9 in Gilbarg-Trudinger's book states that : if $B$ is a ball in $\mathbb{R}^n$ centred at $x_0$ and $f\in C^{\alpha}(B): \sup_{x\in B} (\text{dist}(x, \partial B))^{2-\beta}\vert f(x)\vert ...
2
votes
2answers
115 views

Absolutely Continuous but not Holder continuous

Can some one give me an example of a function which is Absolutely continuous but not Holder continuous? Thanks
3
votes
3answers
121 views

Non Uniformly Elliptic Equations page 117 [G-T]

Let $\Omega\subset\mathbb{R}^n$ be open and bounded. Suppose also that $\Omega$ satisfies the exterior sphere condition at $x_0$ and let $B=B_R(y)$ be a ball such that $B\cap\overline{\Omega}=x_0$. ...
2
votes
1answer
88 views

Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$?

Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$? More precisely, I want to prove that THEOREM. A sequence $\{f_n\}$ is convergent in $C^k(\bar\Omega)$ (or some more ...
1
vote
1answer
85 views

partial derivative of $L^2$ norm?

In the chapter on energy methods for partial differential equations I saw the following: $$\frac{d\|u\|_2^2}{dt}=(u,u_t)+(u_t,u)=\cdots$$ So, why we can't just write ...
4
votes
2answers
124 views

Schwarz Reflection Prinnciple for (Real) Harmonic Functions

Assume $u$ is harmonic in $U^{+}$, $u\equiv0$ on $\partial U^{+}\cap\mathbb{R}^{n}_{+}$, and $u\in\mathscr{C}^{2}(\bar{U}^{+})$, where $U$ is the open ball $B_{1}(0)$ of radius $1$ about the origin in ...
2
votes
1answer
55 views

Hörmander condition extension to the boundary

This is the local property for the solution to be infinitely smooth. However, I can't see the result that relates the regularity inside of the domain and the regularity at the boundary, at least not ...
0
votes
2answers
62 views

An inequality $\| f \|_{L^p} \leq \| f \|_{L^\infty}^{1 - \frac{2}{p}} \| f \|_{L^2}^{\frac{2}{p}}$

What is the name of this inequality $$\| f \|_{L^p(\Bbb R^n)} \leq \| f \|_{L^\infty(\Bbb R^n)}^{1 - \frac{2}{p}} \| f \|_{L^2(\Bbb R^n)}^{\frac{2}{p}}$$ for $p > 2$?And how can I prove this?
0
votes
1answer
295 views

General Solution to Quasilinear PDE using Method of Characteristics

This is a homework that I'm having a bit of trouble with. I posted it previously but there was a typo in my original post. Since I received an answer for the incorrect problem it was suggested that ...
2
votes
1answer
87 views

Is the Sobolev Space $H^k(0,1)$ a banach algebra?

In Adams'book:Sobolev Spaces, I know that if $kp>n,\Omega\subset R^n$ is boundary domain and has cone property, then $W^{k,p}(\Omega)$ could see as a banach algebra. My question is that does it ...
1
vote
1answer
74 views

Laplace equation unit sphere

Laplec Equation is given by $\Delta u=0$ with $\Delta u=\sum_{i=1}^{n}u_{x_ix_i}$ Now I am confused with the following: What are solutions of Laplace Eq. for the inner of the unit sphere, which has ...
1
vote
0answers
48 views

Integration gives inequality

The book I am reading makes a claim I don't understand Suppose that $p(x,\xi) \in C(\bar{\Omega},\mathbb{R})$, and define $P(x,\xi) = \int_0^{\xi}p(x,t)\ dt$. Assume further that there are $\mu > ...
2
votes
1answer
94 views

Show that the pointwise limit of monotone increasing harmonic functions is harmonic

Let $\Omega \subseteq \mathbb{R}^n$ be open and consider a sequence $\{f_k\}_{k \in \mathbb{N}}$, $f \in C^2(\Omega)$ of harmonic functions in $\Omega$ such that $0 \le f_k \le f_{k+1}$ and for which ...
3
votes
0answers
72 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 ...
1
vote
1answer
63 views

Spherical mean of a solution to $\Delta f = |x|^\alpha$

Let $f \in C^2(\mathbb{R}^n)$ be a solution of $\Delta f = |x|^\alpha$, for some $\alpha > 0$. Let $M_f(r) = \frac{1}{\sigma_{n-1}r^{n-1}}\int_{S(r)}f(x)d\sigma(x)$ be the spherical mean of $f$ ...
2
votes
0answers
44 views

what are conormal distributions?

According to the first answer in this post, a conormal distribution $u$ on a manifold $X$ relative to a (closed, embedded) submanifold $Y$ is an element of a Banach (or Hilbert) space $H$ such that ...
2
votes
0answers
53 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 ...
7
votes
1answer
151 views

Is $C_c^{\infty}$ dense in $X_0^{\alpha}$?

While reading papers on fractional Laplacian, I always meet space $X_0^{\alpha}(\mathcal{C}_{\Omega})$ which is defined as following: $$X_0^{\alpha}(\mathcal{C}_{\Omega})=\{z\in ...
0
votes
3answers
223 views

How to prove this partial derivative?

Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider ...
1
vote
3answers
85 views

If $u''>0$ in $\mathbf{R}^+$ then $u$ is unbounded?

If $u$ is a positive function such that $u''>0$ in the whole $\mathbf{R}^+$ then $u$ is unbounded? In fact, I know that if $u''>0$ then $u$ is strictly convex. I think that implies $u$ is ...
0
votes
0answers
55 views

Compact embedding theorem of $W^{k,p}(R^n)$?

Is there some kind of function space $X(R^n)$ which satisfies the compact embedding relation as follows: $W^{k,p}(R^n)\hookrightarrow\hookrightarrow X(R^n)$? Could I guess the indeterminate function ...
1
vote
0answers
100 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 $C^1$).
2
votes
1answer
51 views

Alternate definitions of $C^{1,\alpha}(S^1)$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definitition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
5
votes
2answers
104 views

$ W_0^{1,p}$ norm bounded by norm of Laplacian

Let $f\in W_0^{1,p}(U)$, for $U$ a bounded domain and $p < n/(n-1)$. I am trying to prove that there is an inequality of the form $$\|f\|_{W^{1,p}} \leq C \int_{\Omega} |\Delta f| $$ where ...
4
votes
1answer
78 views

An inequality for $W^{k,p}$ norms

Let $u \in W_0^{2,p}(\Omega)$, for $\Omega$ a bounded subset of $\mathbb R^n$. I am trying to obtain the bound $$\|Du\|_p \leq \epsilon \|D^2 u\|_p + C_\epsilon \|u\|_p$$ for any $\epsilon > 0$ ...
2
votes
2answers
88 views

existence of solution of a degenerate pde with change of variables

I am looking at the pde $$u_t=x^2u_{xx},\; x\in [0,\infty) ,\; t\in (0,T], \; u(0,x)=u_0(x)$$ This is a degenerate pde with a diffusion coefficient which is not bounded from 0, so I can't apply the ...
2
votes
2answers
155 views

How to use the dense theorem to prove this exercise?

Let $\frac{2n}{n+1}\leqslant p<n,\quad q=\frac{np}{2n-p},\quad u\in L^1(R^n)\cap W^{1,p}(R^n)$, then prove $u^2\in W^{1,q}(R^n).$ I hope someone can show me how to prove it by dense theorem ...
1
vote
1answer
61 views

Proving a sequence of distributions converge in $C^{-\infty}(\mathbb{R})$

Question: I have a sequence $(T_n)$, where $T_n$ is given by the locally integrable function $ne^{-\dfrac{n^{2}x^{2}}{2}}$, converges in $C^{-\infty}(\mathbb{R})$ and compute its limit. I suspect ...
1
vote
2answers
67 views

Estimate on a simple-looking integral arising from harmonic analysis/harmonic extensions

Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$: ...
2
votes
1answer
46 views

About the boundedness of the derivative of a function which is in special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, then how can I conclude that $$ \left \| \frac{\partial f}{\partial t} \right \|_{L^\infty([0,T] \times \Bbb R^n )} < \infty ?$$ Here $f$ ...
1
vote
1answer
109 views

Mean Value property for harmonic functions on regions other than balls/spheres

Let $u$ be a harmonic function on $\mathbb R^n$. We know that if $B$ is a ball centered at $x$, then $$u(x) = \frac{1}{|B|} \int_B u(y) dy = \frac{1}{|\partial B|} \int_{\partial B} u(z) dS(z).$$ I ...
0
votes
1answer
67 views

second derivative of solutions to ODE with Lipschitz coefficients

Just the simple ODE with Lipschitz coefficient $a$ \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} We know that the existence and uniqueness holds and the solution is in $C^1$. ...
3
votes
3answers
177 views

Fourier Series: Integral of a Sum or Sum of Integrals?

While touching on Fourier series in a PDEs course, our professor basically waved her hands at the concept that $$ ...
1
vote
1answer
65 views

Convergence of a sequence of solutions

Suppose that $(v_n)$ is a sequence of solutions of $$\begin{cases}-\Delta v_n = f_n&\text{in }\Omega\\ v_n = 0&\text{on }\partial \Omega,\end{cases}$$ where $\Omega \subset \mathbb{R}^2$ is a ...

1 2