# Tagged Questions

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that $||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ... 0answers 126 views ### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space? Preliminary Definition We define the Zygmund spaces$C^r_{*}$with$r>0$,$r \in \mathbb{R}$in the following way: (all the functions are allowed to have values in$\mathbb{R}^m$, via working with ... 1answer 58 views ### Reference: Hardy space regularity of the Jacobian determinant I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes. Theorem: For all$u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ... 0answers 54 views ### About approximate eigenvalue I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4. Suppose$X$is a real Banach Space,$M$is a ... 1answer 105 views ### Interpolation and embeddings for parabolic function spaces I have a somewhat easy looking question on parabolic function spaces: Let$B$be a ball in$\mathbb R^n$and let$T>0$. Denote$Q:=B \times [0,T]$. Assume$f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$... 1answer 130 views ### To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where$\mu(\alpha)$is a non decreasing function ... 1answer 204 views ### Theorem with an example [closed] i have this theorem in the paper they gives an example: but here$H_1$is not satisfied ! How to correct it please? 1answer 111 views ### The class of bounded uniformly continuous functions in viscosity solution theory for Hamilton-Jacobi equations Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class$BUC(\mathbb{R}^n)$or ... 0answers 85 views ### Boundary gradient estimate Assume$U$is the unit disk and$\bar U$its closure and let$u\in C^2(U)\cap C(\bar U)$be a real function, with$u(z)=0$for$z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ... 0answers 113 views ### Bound for a certain integral expression I am working to establish an estimate in$X^{s,b}$spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ... 1answer 182 views ### Inequality in the Sobolev space$H^1$I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ... 1answer 126 views ### Pohozaev result for equations with weights I am interested in nonnegative solutions of -div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p in \Omega with u=0 on \partial \Omega. Or instead the equation -\Delta u + ... 2answers 180 views ### A general inequality about spherical mean of a function suppose \overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1, is the average of u(r,w) on sphere S^{n-1},where (r,w) are the polar coordinates in R^n. My question is ... 0answers 130 views ### Extension of solutions of PDE Let \Omega \subset \mathbb{R}^{2} be an open set such that \mathbf{0} \in \Omega. Let A := \Omega \setminus (\{0\}\times \mathbb{R}), that is, A is \Omega with the y-axis removed. Let ... 1answer 160 views ### Does a particular iteration produce a weak solution to a non linear pde? Consider the following non linear pde in the unknown v(x,y):$$ \frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$where t is some fixed small ... 0answers 144 views ### Is there an appropriate weighted Sobolev space to include exponential map and projection map? Observe that given a non negative function \omega: \mathbb{R^2} \rightarrow [0, \infty), we can define the weighted L^{p}(\mathbb{R}^2, \omega) spaces. They are measurable functions f: ... 0answers 354 views ### Classes of (non-continuous) functions with the fixed point property Let K be a convex body in R^d. (Say, a ball, say a cube...) For which classes \cal C of functions, every function f \in {\cal C} which takes K into itself admits a fixed point in K. ... 1answer 381 views ### Showing a singular integral operator takes Holder continuous functions to Holder continuous functions (of the same order) I would like to show the following function is \gamma-HÃ¶lder continuous. Said function F:\mathbb{R}^n \rightarrow \mathbb{R} is defined by a singular integral operator of convolution type as ... 0answers 179 views ### Viscosity solution of the PDE Let \Omega be bounded domain, u=0 on \delta\Omega and$$|Du|-f(x,u)=0$$where f\ge 0 and f is strictly monotone for fixed x. I am looking for the reference to show that it has unique ... 2answers 199 views ### Alternate definitions of C^{1,\alpha} and C^{1,\alpha}(\bar{D}) maps My question is about the precise definition regarding the following: Let f be an orientation-preserving C^1 diffeomorphism of the unit circle S^1. So f'(b) exists and can be thought as a ... 1answer 148 views ### Subharmonic envelope I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ... 1answer 139 views ### On a limit at the boundary of \mathbb{D} related to complex and harmonic analysis Let p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2} be the Poisson kernel on the open unit disk \mathbb{D}, fix 0<\alpha<1 . Let a\in \partial\mathbb{D}=S^1 be fixed. Then my question is : ... 2answers 152 views ### Decay rate of nonlocal differential operator? Hi, Moers. Let m(\xi) \in S^0, that is,$$ |D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n. $$It's well known that m(D) is bounded in L^p for 1 < p < \infty. ... 1answer 220 views ### Minimizing action squared versus action I have a very basic question in the calculus of variations: Suppose I want to minimize the functional$$A[r, r'] = \int_\Omega L(r, r') dx $$When is it possible to say that extremals of A agree ... 1answer 404 views ### Calculating a distributional derivative Suppose that we have a sequence of functions u_j that are in L^{\infty}(0,1). Then the sequence of maps N_j(s) := \|u_j(s)\|^2 are also in L^{\infty}(0,1). Hence they give rise to ... 2answers 307 views ### Higher order partial derivatives and global regularity. Let f be a function of two variables x and y. Assume that f is C^1. Assume that f_{xx} exists and continuous. Is it true that f_{xy} exists and continuous? Is it true that f_{yx} ... 0answers 232 views ### Density of 0-homogeneous functions in H^1(\partial \Omega) Recall: A function f:\mathbb{R}^n\rightarrow\mathbb{R} is called 0-homogeneous if f(\lambda x)= f(x) for every \lambda>0 and every x\in \mathbb{R}^n. Question: Let B a convex balanced ... 2answers 249 views ### The extension of smooth function If U is a bounded domain in \mathbb R^n whose boundary is smooth, and f is a smooth function on U whose partial derivatives of all orders have a continuous extensions to \bar U. For an ... 2answers 843 views ### Chain rule for fractional laplacian Does anyone know a formula of chain rule for fractional laplacian? say we take the fractional laplacian of order a on function g(U(x)) x\in \mathbb{R}^2, U \in \mathbb{R}, g \colon \mathbb{R} ... 1answer 353 views ### Mean value property with fixed radius Let f be a continuous function defined on \mathbb{R^n}. It is well known that both the spherical mean value property (MVP) of f, i.e.$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ... 2answers 886 views ### Continuation of a smooth function Setting Suppose I have two bounded open domains$\Omega' \subset \Omega \subset \mathbb{R}^n$(I'm particularly interested in case n = 2 or n = 3). We assume that all boundaries of domains are ... 1answer 486 views ### Proof of the “Neo-classical Inequality” I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in$p\geq 1, n$:$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...
Is it true that the characteristic (indicator) function of a subset of Euclidean space with finite positive measure is never in the Sobolev space $H^1 = W^{1,2}$ And if so, what is the ...