Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOptimal conditions for $L^infty$-regularity and a priori estimates for elliptic systems, I: two components
83
0
0.0
(
0
)
Authors
Li Yuxiang
Ask ChatGPT about the research
No Arabic abstract
In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_delta$-solutions. Thanks to the linear theory in $L^p_delta(Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
rate research
Read More
In this paper, we present a bootstrap procedure for general elliptic systems with $n(geq 3)$ components. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_delta$-solutions. Thanks to the linear theory in $L^p_delta(Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
We prove a number of textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Dirichlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability a la {v{S}}ne{ui}berg. Another one is stability of solvability in perturbing the coefficients in $L^infty$ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so.
In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$ |I_alpha F|_{dot{B}^{0,1}_{d/(d-alpha),1}(mathbb{R}^d;mathbb{R}^k)} leq C |F |_{L^1(mathbb{R}^d;mathbb{R}^k)} $$ for all $F in L^1(mathbb{R}^d;mathbb{R}^k)$ which satisfy a first order cocancelling differential constraint. We show how this implies endpoint Besov-Lorentz estimates for Hodge systems with $L^1$ data via fractional integration for exterior derivatives.
This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call emph{balanced cubic estimates}. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru [12], while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard-Burq-Zuily [3, 4], Hunter-Ifrim-Tataru [12] and Ai [2].
Given $Lgeq 1$, we discuss the problem of determining the highest $alpha=alpha(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^alpha_{rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $alpha(L)gtrsim {rm exp}(-CL^beta)$, for some $C, betageq 1$ depending on the dimension $Ngeq 3$. We show that in the non-local case, $alpha(L)gtrsim L^{-1-delta}$ for all $delta>0$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
