Final Report Summary - HAPDEGMT (Harmonic Analysis, Partial Differential Equations and Geometric Measure Theory)
This project lies in the interface of Harmonic Analysis, Partial Differential Equations and Geometric Measure theory. The fruitful interaction between these three fields allows us to tackle problems motivated by elliptic PDEs and require the use of techniques from Harmonic Analysis and Geometric Measure Theory. We deal with good operators (e.g. the Laplacian) in domains that are not topologically friendly or with rough boundaries. Even for nice domains, we treat complex coefficients, which could even be rough, systems or allow the ellipticity to have some degeneracy.
One of the goals has been to extend the F. and M. Riesz theorem to higher dimensions both qualitatively and quantitatively. That result states that harmonic measure is absolutely continuous with respect to arclength measure for simply connected domains in the complex plane with rectifiable boundary. Without assuming strong connectivity, we have shown that if surface measure is absolutely continuous with respect to the harmonic measure in a piece of the boundary with finite surface measure, then that piece has to be rectifiable. For domains with Ahlfors regular boundary we have characterized the solvability of the $L^p$-Dirichlet problem for some finite $p$, or equivalently, the fact that the harmonic measure belongs weak-$A_\infty$. Any of these is equivalent to the uniform recitifiability of the boundary and the weak local John condition, guaranteeing non-tangential local accessibility to portions of the boundary. In the complement of a uniformly rectifiable set, we have developed a powerful Corona decomposition and established that all bounded harmonic functions satisfy Carleson measure estimates and are $\epsilon$-approximable. For 1-sided chord-arc domains (Ahlfors regular boundary, and quantitatively open and connected) we have shown that the boundary is uniformly rectifiable if and only if the domain is chord-arc if and only if the elliptic measure belongs to $A_\infty$ for any elliptic operator in the Kenig-Pipher class. In the same setting, we have studied the perturbation theory: if the discrepancy between two operators satisfies some Carleson measure condition, then the $A_\infty$ property of one elliptic measure is equivalent to the other. Small discrepancies allow us to preserve the corresponding exponent in the reverse Hölder class.
In the upper half-space we have studied boundary value problems for elliptic systems with constant complex coefficients (e.g. the Laplacian or the Lamé system of elasticity) obtaining well-posedness of the Dirichlet problem on very general Köthe function spaces and also in BMO/VMO. As a result we have found classes of dense functions in VMO, improving Sarason’s classical result.
For regular Semmes-Kenig-Toro domains, where small oscillation of the outer unit normal is assumed, and for elliptic systems with constant complex constant coefficients, we have settle the question of the well-posedness of boundary value problems (Dirichlet, Regularity, Neumann, Transmission). For bounded domains, we have linked the well-posedness with the nature of the outward unit normal exhibited through its multiplier properties and oscillatory behavior. In the unbounded setting, we have developed an analogous theory where the outer unit normal has small oscillation in the sense that its BMO seminorm is small.
For second order divergence form degenerate elliptic operators with complex coefficients we have obtained $L^p$-Kato estimates and solved the $L^2$-Kato problem. Both for uniformly and degenerate elliptic operators we have established the solvability of the regularity problem in weighted spaces using our estimates for the conical and vertical square functions. Finally, we have provide several characterizations for the weighted Hardy spaces associated with uniformly elliptic operators.
One of the goals has been to extend the F. and M. Riesz theorem to higher dimensions both qualitatively and quantitatively. That result states that harmonic measure is absolutely continuous with respect to arclength measure for simply connected domains in the complex plane with rectifiable boundary. Without assuming strong connectivity, we have shown that if surface measure is absolutely continuous with respect to the harmonic measure in a piece of the boundary with finite surface measure, then that piece has to be rectifiable. For domains with Ahlfors regular boundary we have characterized the solvability of the $L^p$-Dirichlet problem for some finite $p$, or equivalently, the fact that the harmonic measure belongs weak-$A_\infty$. Any of these is equivalent to the uniform recitifiability of the boundary and the weak local John condition, guaranteeing non-tangential local accessibility to portions of the boundary. In the complement of a uniformly rectifiable set, we have developed a powerful Corona decomposition and established that all bounded harmonic functions satisfy Carleson measure estimates and are $\epsilon$-approximable. For 1-sided chord-arc domains (Ahlfors regular boundary, and quantitatively open and connected) we have shown that the boundary is uniformly rectifiable if and only if the domain is chord-arc if and only if the elliptic measure belongs to $A_\infty$ for any elliptic operator in the Kenig-Pipher class. In the same setting, we have studied the perturbation theory: if the discrepancy between two operators satisfies some Carleson measure condition, then the $A_\infty$ property of one elliptic measure is equivalent to the other. Small discrepancies allow us to preserve the corresponding exponent in the reverse Hölder class.
In the upper half-space we have studied boundary value problems for elliptic systems with constant complex coefficients (e.g. the Laplacian or the Lamé system of elasticity) obtaining well-posedness of the Dirichlet problem on very general Köthe function spaces and also in BMO/VMO. As a result we have found classes of dense functions in VMO, improving Sarason’s classical result.
For regular Semmes-Kenig-Toro domains, where small oscillation of the outer unit normal is assumed, and for elliptic systems with constant complex constant coefficients, we have settle the question of the well-posedness of boundary value problems (Dirichlet, Regularity, Neumann, Transmission). For bounded domains, we have linked the well-posedness with the nature of the outward unit normal exhibited through its multiplier properties and oscillatory behavior. In the unbounded setting, we have developed an analogous theory where the outer unit normal has small oscillation in the sense that its BMO seminorm is small.
For second order divergence form degenerate elliptic operators with complex coefficients we have obtained $L^p$-Kato estimates and solved the $L^2$-Kato problem. Both for uniformly and degenerate elliptic operators we have established the solvability of the regularity problem in weighted spaces using our estimates for the conical and vertical square functions. Finally, we have provide several characterizations for the weighted Hardy spaces associated with uniformly elliptic operators.