# Tagged Questions

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### 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 ...
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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 ...
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### 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 ...
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I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ... 1answer 113 views ### Pohozaev result for equations with weights I am interested in nonnegative solutions of -div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p in \Omega with  u=0 on  \partial \Omega. Or instead the equation  -\Delta u + ... 2answers 166 views ### A general inequality about spherical mean of a function suppose \overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1, is the average of u(r,w) on sphere S^{n-1},where (r,w) are the polar coordinates in R^n. My question is ... 0answers 124 views ### Extension of solutions of PDE Let \Omega \subset \mathbb{R}^{2} be an open set such that \mathbf{0} \in \Omega. Let A := \Omega \setminus (\{0\}\times \mathbb{R}), that is, A is \Omega with the y-axis removed. Let ... 1answer 158 views ### Does a particular iteration produce a weak solution to a non linear pde? Consider the following non linear pde in the unknown v(x,y):$$ \frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$where t is some fixed small ... 0answers 130 views ### Is there an appropriate weighted Sobolev space to include exponential map and projection map? Observe that given a non negative function \omega: \mathbb{R^2} \rightarrow [0, \infty), we can define the weighted L^{p}(\mathbb{R}^2, \omega)  spaces. They are measurable functions f: ... 0answers 334 views ### Classes of (non-continuous) functions with the fixed point property Let K be a convex body in  R^d. (Say, a ball, say a cube...) For which classes  \cal C of functions, every function  f \in {\cal C} which takes K into itself admits a fixed point in K. ... 1answer 353 views ### Showing a singular integral operator takes Holder continuous functions to Holder continuous functions (of the same order) I would like to show the following function is \gamma-HÃ¶lder continuous. Said function F:\mathbb{R}^n \rightarrow \mathbb{R} is defined by a singular integral operator of convolution type as ... 0answers 169 views ### Viscosity solution of the PDE Let \Omega be bounded domain, u=0 on \delta\Omega and$$|Du|-f(x,u)=0$$where f\ge 0 and f is strictly monotone for fixed x. I am looking for the reference to show that it has unique ... 2answers 194 views ### Alternate definitions of C^{1,\alpha} and C^{1,\alpha}(\bar{D}) maps My question is about the precise definition regarding the following: Let f be an orientation-preserving C^1 diffeomorphism of the unit circle S^1. So f'(b) exists and can be thought as a ... 1answer 146 views ### Subharmonic envelope I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ... 1answer 134 views ### On a limit at the boundary of \mathbb{D} related to complex and harmonic analysis Let p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2} be the Poisson kernel on the open unit disk \mathbb{D}, fix 0<\alpha<1 . Let a\in \partial\mathbb{D}=S^1 be fixed. Then my question is : ... 2answers 151 views ### Decay rate of nonlocal differential operator? Hi, Moers. Let m(\xi) \in S^0, that is,$$ |D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n. $$It's well known that m(D) is bounded in L^p for 1 < p < \infty. ... 1answer 219 views ### Minimizing action squared versus action I have a very basic question in the calculus of variations: Suppose I want to minimize the functional$$A[r, r'] = \int_\Omega L(r, r') dx $$When is it possible to say that extremals of A agree ... 1answer 376 views ### Calculating a distributional derivative Suppose that we have a sequence of functions u_j that are in L^{\infty}(0,1). Then the sequence of maps N_j(s) := \|u_j(s)\|^2 are also in L^{\infty}(0,1). Hence they give rise to ... 2answers 298 views ### Higher order partial derivatives and global regularity. Let f be a function of two variables x and y. Assume that f is C^1. Assume that f_{xx} exists and continuous. Is it true that f_{xy} exists and continuous? Is it true that f_{yx} ... 0answers 225 views ### Density of 0-homogeneous functions in H^1(\partial \Omega) Recall: A function f:\mathbb{R}^n\rightarrow\mathbb{R} is called 0-homogeneous if f(\lambda x)= f(x) for every \lambda>0 and every x\in \mathbb{R}^n. Question: Let B a convex balanced ... 2answers 224 views ### The extension of smooth function If U is a bounded domain in \mathbb R^n whose boundary is smooth, and f is a smooth function on U whose partial derivatives of all orders have a continuous extensions to \bar U. For an ... 2answers 778 views ### Chain rule for fractional laplacian Does anyone know a formula of chain rule for fractional laplacian? say we take the fractional laplacian of order a on function g(U(x)) x\in \mathbb{R}^2, U \in \mathbb{R}, g \colon \mathbb{R} ... 1answer 331 views ### Mean value property with fixed radius Let f be a continuous function defined on \mathbb{R^n}. It is well known that both the spherical mean value property (MVP) of f, i.e.$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
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 ...