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We consider a nonlinear critical problem involving the fractional Laplacian operator arising in conformal geometry, namely the prescribed -curvature problem on the standard -sphere, . Under the assumption that the prescribed function is flat near its critical points, we give precise estimates on the losses of the compactness and we provide existence results. In this first part, we will focus on the case , which is not covered by the method of Jin, Li, and Xiong (2014, 2015).
We study qualitative positivity properties of quasilinear equations of the form
where is a domain in , , is a symmetric and locally uniformly positive definite matrix, is a real potential in a certain local Morrey space (depending on ), and
Our assumptions on the coefficients of the operator for are the minimal (in the Morrey scale) that ensure the validity of the local Harnack inequality and hence the Hölder continuity of the solutions. For some of the results of the paper we need slightly stronger assumptions when .
We prove an Allegretto–Piepenbrink-type theorem for the operator , and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case , we examine the behaviour of a positive solution near a nonremovable isolated singularity and characterize the existence of the positive minimal Green function for the operator in .
In this article we are interested in the rigorous construction of geometric optics expansions for hyperbolic corner problems. More precisely we focus on the case where self-interacting phases occur. Those phases are proper to the high frequency asymptotics for the corner problem and correspond to rays that can display a homothetic pattern after a suitable number of reflections on the boundary. To construct the geometric optics expansions in that framework, it is necessary to solve a new amplitude equation in view of initializing the resolution of the WKB cascade.
We make use of the flexibility of infinite-index solutions to the Allen–Cahn equation to show that, given any compact hypersurface of with , there is a bounded entire solution of the Allen–Cahn equation on whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of . More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.
has the interesting feature that an associated Rayleigh quotient is nonincreasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hölder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional -Laplacian.
We provide a different proof for the global regularity of potential functions in the optimal transport problem, which was originally proved by Caffarelli. Our method applies to a more general class of domains.
For and , let be the fractional differential operator and be the singular integral operator. We obtain a necessary and sufficient condition on the function to guarantee that is a bounded operator on a function space such as and for any . Furthermore, we establish a necessary and sufficient condition on the function to guarantee that is a bounded operator from to and from to . This is a new theory. Finally, we apply our general theory to the Hilbert and Riesz transforms.