On the absolutely continuous spectrum of generalized indefinite strings II

We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-\infty,-1/2]\cup [1/2,\infty)$.Comment: 27 page

Higher regularity of free boundaries in obstacle problems

[eng] In the thesis we consider higher regularity of the free boundaries in different variations of the obstacle problem, that is, when the Laplace operator b. is replaced with another elliptic or parabolic operator. In the fractional obstacle problem with drift (L = (-'6.)8 + b · v'), we prove that for constant b, and irrational s > ½ the free boundary is C00 near regular points as long as the obstacle is C00. To do so we establish higher order boundary Harnack inequalities for linear equations. This gives a bootstrap argument, as the normal of the free boundary can be expressed with quotients of derivatives of solution to the obstacle problem. Furthermore we establish the boundary Harnack estímate for linear parabolic operators (L = Ot - b.) in parabolic C1 and C1•°' domains and give a new proof of the higher order boundary Harnack estímate in ck,a domains. In the similar way as in the fractional obstacle problem with drift this implies that the free boundary in the parabolic obstacle problem is C00 near regular points. We also study the regularity of the free boundary in the parabolic fractional obstacle problem (L = Ot + (-b.)8) in the cases > ½- We are able to provea boundary Harnack estímate in C1•°' domains, which improves the regularity of the free boundary from C1•°' to C2•°'. Finally, we establish the full regularity theory for free boundaries in fully non-linear parabolic obstacle problem. Concretely we find the splitting of the free boundary into regular and singular points, we show that near regular points the free boundary is locally a graph of a C00 function, and that the singular points are '' rare" - they can be covered with a Lipschitz manifold of co-dimension 2, which is arbitrarily flat in space

On the absolutely continuous spectrum of generalized indefinite strings

V delu raziskujemo absolutno zvezni del spektra posplošenih strun. Pokažemo, da se absolutno zvezni spekter nekaterih modelnih primerov posplošenih strun ohrani pod ustreznimi perturbacijami. V prvem delu razvijemo teorijo in orodja, potrebna za definicijo in izračun spektra. V drugem delu predstavimo rezultate in jih dokažemo.We investigate the absolutely continuous spectrum of generalized indefinite strings. We show that the absolutely continuous spectrum of two model examples of generalized indefinite strings is preserved under rather wide perturbations. In the first part of the thesis we present the theory and tools needed for the definition and computation of the spectrum. In the second part we present the results and their proofs

Regularity theory for fully nonlinear parabolic obstacle problems

We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is C∞ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension n−1 which is also ε-flat in space, for any ε>0

On the absolutely continuous spectrum of generalized indefinite strings II

We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set   (−∞, −1/2] ∪ [1/2,∞).</p