Quasiregular semigroups.
Mappings of finite distortion: Compactness.
Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions
CONTENTSPreliminaries........................................................................................................ 51. Auxiliary results......................................................................................................... 132. The second order equations.................................................................................. 143. Some properties of Sobolev and Besov spaces................................................ 204. Classes , 0 < a ≤ 1...............................................................................
Nonlinear Commutators and Jacobians.
Nonlinear analysis and quasiconformal mappings from the perspective of PDEs
Contents Introduction 119 1. Quasiregular mappings 120 2. The Beltrami equation 121 3. The Beltrami-Dirac equation 122 4. A quest for compactness 124 5. Sharp -estimates versus variational integrals 125 6. Very weak solutions 128 7. Nonlinear commutators 129 8. Jacobians and wedge products 131 9. Degree formulas 134 References 136
Projections onto gradient fields and -estimates for degenerated elliptic operators
An essay on the Interpolation Theorem of Józef Marcinkiewicz - Polish patriot
Riesz transforms and related singular integrals.
Interpolation theorem for the p-harmonic transform
We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation . In this example the p-harmonic transform is essentially inverse to . To every vector field our operator assigns the gradient of the solution, . The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments...
Squeezing the Sierpinski sponge
We give an example relating to the regularity properties of mappings with finite distortion. This example suggests conditions to be imposed on the distortion function in order to avoid "cavitation in measure".
Quasiharmonic fields
Caccioppoli estimates and very weak solutions of elliptic equations
Caccioppoli estimates are instrumental in virtually all analytic aspects of the theory of partial differential equations, linear and nonlinear. And there is always something new to add to these estimates. We emphasize the fundamental role of the natural domain of definition of a given differential operator and the associated weak solutions. However, we depart from this usual setting (energy estimates) and move into the realm of the so-called very weak solutions where important new applications lie....
New and old function spaces in the theory of PDEs and nonlinear analysis
Regularity of p-harmonic functions on the plane.
Divergence forms of the infinity-Laplacian.
The central theme running through our investigation is the infinity-Laplacian operator in the plane. Upon multiplication by a suitable function we express it in divergence form, this allows us to speak of weak infinity-harmonic function in W1,2. To every infinity-harmonic function u we associate its conjugate function v. We focus our attention to the first order Beltrami type equation for h= u + iv
On the Product of Functions in and
The point-wise product of a function of bounded mean oscillation with a function of the Hardy space is not locally integrable in general. However, in view of the duality between and , we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic...
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