Opuscula Mathematica

Galerkin-type minimizers to a competing problem for (\vec{p},\vec{q})-Laplacian with variable exponents

Tue, 27 Jan 2026 11:00:05 +0100

Author(s): Zhenfeng Zhang, Mina Ghasemi, Calogero Vetro.

Abstract: This study focuses on a sequence of approximate minimizers for the functional \[J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,\] where \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain, and \(p_i,q_i\in C(\overline{\Omega})\) with \(1\lt p_i,q_i\lt +\infty\) for all \(i \in \{1,\ldots,N\}\). We establish the convergence result to the infimum of \(J(u)\) when \(F:\mathbb{R}\to\mathbb{R}\) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional.
Keywords: anisotropic Sobolev space, Clarke generalized gradient, Dirichlet problem, Galerkin basis, \((\vec{p},\vec{q})\)-Laplacian with variable exponents.
Mathematics Subject Classification: 46E30, 47J22.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 101-119, https://doi.org/10.7494/OpMath.202511231.

Calderón-Hardy type spaces and the Heisenberg sub-Laplacian

Tue, 27 Jan 2026 11:00:04 +0100

Author(s): Pablo Rocha.

Abstract: For \(0 \lt p \leq 1 \lt q \lt \infty\) and \(\gamma \gt 0\), we introduce the Calderón-Hardy spaces \(\mathcal{H}^{p}_{q,\gamma}(\mathbb{H}^{n})\) on the Heisenberg group \(\mathbb{H}^{n}\), and show for every \(f \in H^{p}(\mathbb{H}^{n})\) that the equation \[\mathcal{L}F=f\] has a unique solution \(F\) in \(\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})\), where \(\mathcal{L}\) is the sub-Laplacian on \(\mathbb{H}^{n}\), \[1 \lt q \lt \frac{n+1}{n} \quad \text{and} \quad (2n+2)\left(2+\frac{2n+2}{q}\right)^{-1} \lt p \leq 1.\]
Keywords: Calderón-Hardy type spaces, Hardy type spaces, atomic decomposition, Heisenberg group, sub-Laplacian.
Mathematics Subject Classification: 42B25, 42B30, 42B35, 43A80.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 73-99, https://doi.org/10.7494/OpMath.202512221.

A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms

Tue, 27 Jan 2026 11:00:03 +0100

Author(s): Ahmed Mohammed, Giovanni Porru.

Abstract: In this short note we study a Harnack inequality for \(k\)-Hessian equations that involve nonlinear lower-order terms which depend on the solution and its gradient.
Keywords: \(k\)-Hessian equation, \(k\)-convex solutions, Harnack inequality, Liouville property.
Mathematics Subject Classification: 35J60, 35J70, 35B45, 35B53.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 55-72, https://doi.org/10.7494/OpMath.202512261.

Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller

Tue, 27 Jan 2026 11:00:02 +0100

Author(s): Kazuki Ishibashi.

Abstract: This study investigated the asymptotic behavior of linear differential systems incorporating a proportional derivative-type (PD) differential operator. Building on the classical asymptotic convergence property of Wintner, a generalized Wintner-type asymptotic result was established for such systems. The proposed framework encompasses a wide class of time-varying coefficient matrices and extends classical asymptotic theory to equations governed by PD operators. An illustrative example is presented to demonstrate the applicability of the proposed theorem.
Keywords: Wintner-type, asymptotic behavior, linear differential systems, proportional-derivative controller.
Mathematics Subject Classification: 34D05, 26A24.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 41-54, https://doi.org/10.7494/OpMath.202512101.

Parametric formal Gevrey asymptotic expansions in two complex time variable problems

Tue, 27 Jan 2026 11:00:01 +0100

Author(s): Guoting Chen, Alberto Lastra, Stéphane Malek.

Abstract: The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution.
Keywords: singularly perturbed, formal solution, several complex variables, Cauchy problem.
Mathematics Subject Classification: 35C10, 35R10, 35C15, 35C20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271.