Hölder Continuity and Boundedness Estimates for Nonlinear Fractional Equations in the Heisenberg Group
Maria Manfredini, Giampiero Palatucci, Mirco Piccinini, Sergio Polidoro
Abstract
Abstract We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p -Laplacian operator on the Heisenberg-Weyl group $$\mathbb {H}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> . Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.
Topics & Concepts
Heisenberg groupMathematicsDifferential geometryNonlinear systemGroup (periodic table)Fourier analysisMathematical analysisFractional calculusFourier transformPure mathematicsPhysicsQuantum mechanicsNonlinear Partial Differential EquationsAdvanced Mathematical Physics ProblemsStability and Controllability of Differential Equations