Title: An L∞ Estimate for Solutions to p-Laplacian Type Equations Using an Obstacle Approach and Applications

Abstract: In this paper, we address an L∞ estimate for weak solutions of p-Laplacian type equations of the form

−div a(x, ∇u) = f(x), in Ω,

where 1 < p < ∞, Ω ⊂ Rⁿ (n ≥ 2) is a suitable domain, possibly unbounded but with finite measure, and f ∈ L^q(Ω) with q > n/p and q ≥ p/(p − 1). The mapping a : Ω × Rⁿ → Rⁿ is a continuous vector field satisfying appropriate structural properties.

Unlike the non-variational setting, our approach is based on the analysis of an obstacle problem associated with the equation and on the investigation of its intrinsic qualitative properties. Moreover, in certain scenarios, our estimates reveal new geometric quantities that are absent from the classical literature. Finally, we present some applications of these estimates, which may be essential in specific contexts of quasilinear partial differential equations and related topics.

Title: A Fully Nonlinear Degenerate Free Transmission Problem

Abstract: We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of a viscosity solution to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities, we prove regularity results for the constructed viscosity solution to the problem. Our findings include regularity in C1,α spaces, and an explicit characterization of α in terms of the degeneracy rates. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation.

Title: The inhomogeneous p-obstacle problem with Robin boundary condition

Abstract: In this talk we discuss a inhomogeneous obstacle problem governed by the p-Laplacian with Robin boundary condition. Our findings include existence of weak solutions as well as uniform estimates in the parameter p in a suitable functional space.  Moreover, we characterize the limit profile as a viscosity solution of a obstacle problem with some boundary condition.  

This is a joint-work with J.V. da Silva (IMECC-Unicamp) and R.T. Frias (IFSP) .

Title: Rigidity for Non-Uniformly Elliptic Equations in Planar Strip Domains

Abstract: We consider smooth solutions to a possibly non-uniformly elliptic equation of the form
Δ_g u = 0 in Ω,

where the domain Ω ⊂ R² is unbounded and bounded by two graphs J₀ and J₁.
The function u satisfies mixed boundary conditions:

u = 0 and ∂u/∂η = c₀ on J₀,
u = 1 and ∂u/∂η = c₁ on J₁,

with c₀, c₁ being constants.
Under the assumptions 0 < u < 1 in Ω and J₀, J₁ ∈ C¹,¹(R), we prove that the only possible configuration is the trivial one: u must be linear, and the boundary graphs J₀ and J₁ are parallel straight lines. The argument relies on the comparison principle and highlights a rigidity phenomenon for this class of elliptic equations.

Joint work with David Jesus (Univ. Bologna), Diego Moreira (UFC), and Sergio H. Monari Soares (ICMC/USP).

Title: Lipschitz Regularity of Vectorial Almost-Minimizers for Alt–Caffarelli Functionals in Orlicz–Sobolev Spaces

Abstract: For a fixed constant λ > 0 and a bounded Lipschitz domain Ω ⊂ ℝⁿ, with n ≥ 2, we show that almost-minimizers (functions satisfying a suitable variational inequality) of the Alt–Caffarelli type functional

J_G (v; Ω) = ∫Ω ( ∑{i=1}^m G(|∇vᵢ(x)|) + λ χ_{|v|>0}(x) ) dx,

where v = (v₁, …, v_m) and m ∈ ℕ, exhibit optimal Lipschitz continuity on compact subsets of Ω. Here, G is a Young function satisfying appropriate growth conditions.

Moreover, we obtain universal gradient estimates for non-negative almost-minimizers in the interior of the non-coincidence set. Our results extend recent regularity theory for weakly coupled vectorial almost-minimizers for the p-Laplacian, as studied by Bayrami et al., as well as the scalar case treated by da Silva et al., Dipierro et al., and Pellegrino–Teixeira. This provides new insights and techniques applicable to a broad class of nonlinear one- and two-phase free boundary problems with non-standard growth.

This is joint work with Pedro Pontes (Jilin University, China) and Minbo Yang (Zhejiang Normal University, China).

Title: Improved Regularity for Fully Nonlinear Elliptic Equations in Borderline Cases

Abstract: We improve the regularity of solutions to fully nonlinear elliptic equations of the form F(D²u, x) = f, based on the weakest and borderline integrability properties of the source term f at points where the solution is touched from below by sufficiently smooth functions. In particular, we show that solutions exhibit higher regularity at local minima.

Title: On the asymptotically linear problem for an elliptic equation with an indefinite nonlinearity

Abstract: Let Ω be an open bounded subset of Rⁿ. We consider the problems

(1) −Δu = Λ Q_Ω(x)u, u ∈ H¹(Rⁿ) \ {0},

(2) −Δv = Q_Ω |v|^{p−1}v, v ∈ D^{1,p}(Rⁿ) ∩ Lᵖ(Rⁿ),

where Q_Ω ≡ 1 in Ω and Q_Ω ≡ −1 in Rⁿ \ Ω, with n ≥ 3.

In this talk, we show the existence of a discrete spectrum on the positive real line, as well as the classical Faber–Krahn inequality and the nonlocal Hong–Krahn–Szegő inequality related to optimization problems for the first and second eigenvalues, respectively. We further analyze properties of the associated eigenfunctions, including their asymptotic decay and nodal domains. In addition, we prove that the positive solution to problem (2) converges to a first positive eigenfunction of problem (1) as p → 2.

Title: On a nonlocal minimization problem with free boundary

Abstract: In this work, we study a free boundary problem associated with the minimization of a nonlocal functional with a volume penalization, namely the Bernoulli problem for the fractional p-Laplacian. We obtain the existence of minimizers, as well as some initial qualitative properties. Moreover, we establish local Hölder regularity and a possible optimal growth at the free boundary.