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In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary., To appear in Mathematics in Engineering, special issue in honor of Sandro Salsa

We establish Weiss’ and Monneau’s type quasi-monotonicity formulas for quadratic energies having matrix of coefficients in a Sobolev space with summability exponent larger than the space dimension and provide an application to the corresponding free boundary analysis for the related classical obstacle problems.

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Holder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999).

This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including L a = div ( | y | a ∇ ) , with a ∈ ( − 1 , 1 ) and their perturbations. As they belong to the Muckenhoupt class A 2 , these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni [1] , [2] , [3] and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension [4] . Our goal in the present paper is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin [5] , [6] , [7] .

The contents of this thesis are an assortment of results in analysis and subRiemannian geometry, with a special focus on the Heisenberg group Hn, Heisenbergtype (H-type) groups, and Carnot groups. As we wish for this thesis to be relatively self-contained, the main definitions and background are covered in Chapter 1. This includes basic information about Carnot groups, Hn, H-type groups, diffusion operators, and the curvature dimension inequality. Chapter 2 incorporates excerpts from a paper by N. Garofalo and the author, [42]. In it, we propose a generalization of Almgren’s frequency function N : (0, 1) → R for solutions to the sub-elliptic Laplace equation ΔHu = 0 in the unit ball of a Carnot group of arbitrary step. If the function u has vanishing discrepancy, then the frequency is monotonically non-decreasing, and we are able to prove a form of strong unique continuation for such functions. Chapter 3 grew out of the author seeking parabolic montonicity formulas in the same vein as Almgren’s frequency. These include two types of monotonicity formulas, those of Struwe- and Poon-type [72], [67]. If a diffusion operator L on a complete manifold M satisfies the curvature dimension inequality CD(ρ, n), then we are able to prove that for solutions to L u = ut in M × (0, T), Struwe’s energy monotonicity holds, at least for time values close enough to T. We introduce a new condition, C(ω) where ω ∈ C1(0, T), related to the Hessian of the heat kernel, and are able to prove a Poon-type frequency monotonicity formula when taking into account a weighting factor depending on ω. We also give examples of manifolds satisfying C(ω), the most interesting of which includes the Ornstein-Uhlenbeck operator. Monotonicity of the weighted frequency also implies a form of strong-unique continuation. In Chapter 4, we derive asymptotics for the heat kernel on H-type groups and generalize a gradient bound from a paper of Garofalo and Segala [43] to these groups. This gradient bound in turn implies a strong Harnack inequality and Wiener criterion similar to those found in [31] and [43].

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely ∫ Ω | ∇ u ( x ) | 2 d x + Per σ ( { u > 0 } , Ω ) , with σ ∈ ( 0 , 1 ) . We obtain regularity results for the minimizers and for their free boundaries ∂ { u > 0 } using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.