Rendiconti di Matematica di Sapienza Università di Roma

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ISSN 1120-7183 (print) ISSN 2532-3350 (online) Home Editorial Board Latest articles Published Volumes Historical Notes Editorial Policy Publication Ethics Instructions for Authors Copyright Subscription Information Editorial Office rendmat@mat.uniroma1.it	Latest Articles Latest accepted (not yet published) papers, forthcoming into regular issues of the journal Once a paper is accepted it goes immediately into production. It is a policy of the journal to make papers available online within four working weeks of acceptance (whenever the authors are collaborative). Denys Dutykh, Éric Leichtnam On complex algebraic singularities of some genuinely nonlinear PDEs 📥 Full Text (PDF format), posted online 28 August 2024. Abstract. In this manuscript, we highlight a new phenomenon of complex algebraic singularities formation for solutions of a large class of genuinely nonlinear Partial Differential Equations (PDEs). We start from a unique Cauchy datum which is holomorphic ramified like x₁^{1/(k + 1)} (and its powers) around the smooth locus x₁ = 0 and is sufficiently singular. Then, we expect the existence of a solution which should be holomorphic ramified around the singular locus S defined by the vanishing of the discriminant of an algebraic equation of degree k + 1. Notice, moreover, that the monodromy of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian and that S depends on the Cauchy datum in contrast to the Leray principle (stated for linear problems only). This phenomenon is due to the fact that the PDE is genuinely nonlinear and that the Cauchy datum is sufficiently singular. First, we investigate the case of the inviscid Burgers Equation (iBE). Later, we state a general Conjecture 9.2, which describes the expected phenomenon. We view this Conjecture 9.2 as a working programme allowing us to develop interesting new Mathematics. We also state Conjecture 7.1, which is a particular case of the general Conjecture 9.2 but keeps all the flavour and difficulty of the subject. Then, we propose a new algorithm with a map F such that a fixed point of F would give a solution to the problem associated with Conjecture 7.1. Then, we perform convincing, elaborate numerical tests which suggest that a Banach norm should exist for which the mapping F should be a contraction so that the solution (with the above specific algebraic structure) should be unique. Samuel Ssekajja Non-existence of certain lightlike hypersurfaces of an indefinite Sasakian manifold 📥 Full Text (PDF format), posted online 24 August 2024. Abstract. We consider a lightlike hypersurface, tangent to the structure vector field, of an indefinite Sasakian manifold. We prove that no such a hypersurface can either have parallel or recurrent second fundamental forms. In addition to the above, we also prove that no such a hypersurface may have parallel or recurrent induced structural tensors. Loïc Foissy, Claudia Malvenuto The Fortuin-Kasteleyn polynomial as a bialgebra morphism and applications to the Tutte polynomial 📥 Full Text (PDF format), posted online 24 August 2024. Abstract. We compute an explicit formula for the antipode of the double bialgebra of graphs in terms of totally acyclic partial orientations, using some general results on double bialgebras. In analogy to what was already proven in Hopf-algebraic terms for the chromatic polynomial of a graph, we show that the Fortuin-Kasteleyn polynomial (a variant of the Tutte polynomial) is a morphism of the double algebra of graphs into that of polynomials, which generalizes the chromatic polynomial. When specialized at particular values, we give combinatorial interpretations of the Tutte polynomial of a graph, via covering graphs and covering forests, and of the Fortuin-Kasteleyn polynomial, via pairs of vertex-edge colorings. Finally we show that the map associating to a graph all its orientations is a Hopf morphism from the double bialgebra of graphs into the one of oriented graphs, allowing to give interpretations of the Fortuin-Kasteleyn polynomial when computed at negative values.. Mohamed Badr Benboubker, Rajae Bentahar, Meryem El Lekhlifi, Hassane Hjiaj Existence of renormalized solutions for some non-coercive anisotropic elliptic problems with Neumann boundary condition 📥 Full Text (PDF format), posted online 31 May 2024. Abstract. Not available in HTML format. Luca Rizzi, Francesco Zucconi Global Kodaira-Spencer class and Massey products 📥 Full Text (PDF format), posted online 27 May 2024. Abstract. We define a new notion of supported global deformation class for a semistable family of complex varieties over a curve f:X→B. We use this notion to study when X, possibly up to a finite covering, has a generically finite morphism onto a product B ×Y with Y of general type. Tilak Bhattacharya Some results for the Asymptotics and the Strong Minimum Principle for solutions to some nonlinear parabolic equations 📥 Full Text (PDF format), posted online 17 May 2024. Abstract. We extend some results by Bhattacharya and Marazzi on strong minimum principle and asymptotics of positive viscosity solutions to a class of doubly nonlinear parabolic equations. Hùng Việt Chu Strong partially greedy bases with respect to an arbitrary sequence 📥 Full Text (PDF format), postedd online 13 May 2024. Abstract. For Schauder bases, Dilworth et al. introduced and characterized the partially greedy property, which is strictly weaker than the (almost) greedy property. Later, Berasategui et al. defined and studied the strong partially greedy property for general bases. Let n be any strictly increasing sequence of positive integers. In this paper, we define the strong partially greedy property with respect to n, called the (n, strong partially greedy) property. We give characterizations of this new property, study relations among (n, strong partially greedy) properties for different sequences n, establish Lebesgue-type inequalities for the (n, strong partially greedy) parameter, investigate (n, strong partially greedy) bases with gaps, and weighted (n, strong partially greedy) bases, to name a few. Furthermore, we introduce the (n, almost greedy) property and equate the property to a strengthening of the (n, strong partially greedy) property. This paper can be viewed both as a survey of recent results regarding strong partially greedy bases and as an extension of these results to an arbitrary sequence instead of N. Lucio Cadeddu, Antonio Greco, Benyam Mebrate Non-autonomous overdetermined problems for the normalized p-Laplacian 📥 Full Text (PDF format), posted online 27 March 2024. Abstract. We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition at the boundary. The definitions and statements needed to understand the main results are recalled in detail. Lucio Boccardo, Andrea Dall'Aglio Bounded solutions for Dirichlet problems with degenerate coercivity and a quadratic gradient term 📥 Full Text (PDF format), posted online 27 March 2024. Abstract. We give existence results for weak solutions of Dirichlet problems for elliptic equations having degenerate coercivity and a first order term which has quadratic growth with respect to the gradient. The proof is based on the use of test functions having exponential growth. Fatima Ezzahra Bourhim, Ali El Mfadel, M'hamed Elomari, Naoufel Hatime Existence and uniqueness results for nonlinear hybrid Ψ-Caputo-type fractional differential equations with nonlocal periodic boundary conditions 📥 Full Text (PDF format), posted online 21 February 2024. Abstract. In this paper, we consider a nonlinear fractional hybrid differential equation involving the Ψ-Caputo fractional operator with nonlocal periodic boundary conditions. Based on Lipschitz and Carathéodory conditions and via the Krasnoselskii fixed point theorem and some basic fractional analysis techniques, we discuss the existence and uniqueness of solutions to the proposed problem. Moreover, for a specific class of continuous functions, we prove some fundamental fractional differential inequalities. We finish this work with a non-trivial example.

Latest Articles

Latest accepted (not yet published) papers, forthcoming into regular issues of the journal

Once a paper is accepted it goes immediately into production. It is a policy of the journal to make papers available online within four working weeks of acceptance (whenever the authors are collaborative).

Denys Dutykh, Éric Leichtnam

On complex algebraic singularities of some genuinely nonlinear PDEs

📥 Full Text (PDF format), posted online 28 August 2024.

Abstract. In this manuscript, we highlight a new phenomenon of complex algebraic singularities formation for solutions of a large class of genuinely nonlinear Partial Differential Equations (PDEs). We start from a unique Cauchy datum which is holomorphic ramified like x₁^{1/(k + 1)} (and its powers) around the smooth locus x₁ = 0 and is sufficiently singular. Then, we expect the existence of a solution which should be holomorphic ramified around the singular locus S defined by the vanishing of the discriminant of an algebraic equation of degree k + 1. Notice, moreover, that the monodromy of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian and that S depends on the Cauchy datum in contrast to the Leray principle (stated for linear problems only). This phenomenon is due to the fact that the PDE is genuinely nonlinear and that the Cauchy datum is sufficiently singular. First, we investigate the case of the inviscid Burgers Equation (iBE). Later, we state a general Conjecture 9.2, which describes the expected phenomenon. We view this Conjecture 9.2 as a working programme allowing us to develop interesting new Mathematics. We also state Conjecture 7.1, which is a particular case of the general Conjecture 9.2 but keeps all the flavour and difficulty of the subject. Then, we propose a new algorithm with a map F such that a fixed point of F would give a solution to the problem associated with Conjecture 7.1. Then, we perform convincing, elaborate numerical tests which suggest that a Banach norm should exist for which the mapping F should be a contraction so that the solution (with the above specific algebraic structure) should be unique.

Samuel Ssekajja

Non-existence of certain lightlike hypersurfaces of an indefinite Sasakian manifold

📥 Full Text (PDF format), posted online 24 August 2024.

Abstract. We consider a lightlike hypersurface, tangent to the structure vector field, of an indefinite Sasakian manifold. We prove that no such a hypersurface can either have parallel or recurrent second fundamental forms. In addition to the above, we also prove that no such a hypersurface may have parallel or recurrent induced structural tensors.

Loïc Foissy, Claudia Malvenuto

The Fortuin-Kasteleyn polynomial as a bialgebra morphism and applications to the Tutte polynomial

📥 Full Text (PDF format), posted online 24 August 2024.

Abstract. We compute an explicit formula for the antipode of the double bialgebra of graphs in terms of totally acyclic partial orientations, using some general results on double bialgebras. In analogy to what was already proven in Hopf-algebraic terms for the chromatic polynomial of a graph, we show that the Fortuin-Kasteleyn polynomial (a variant of the Tutte polynomial) is a morphism of the double algebra of graphs into that of polynomials, which generalizes the chromatic polynomial. When specialized at particular values, we give combinatorial interpretations of the Tutte polynomial of a graph, via covering graphs and covering forests, and of the Fortuin-Kasteleyn polynomial, via pairs of vertex-edge colorings. Finally we show that the map associating to a graph all its orientations is a Hopf morphism from the double bialgebra of graphs into the one of oriented graphs, allowing to give interpretations of the Fortuin-Kasteleyn polynomial when computed at negative values..

Mohamed Badr Benboubker, Rajae Bentahar, Meryem El Lekhlifi, Hassane Hjiaj

Existence of renormalized solutions for some non-coercive anisotropic elliptic problems with Neumann boundary condition

📥 Full Text (PDF format), posted online 31 May 2024.

Abstract. Not available in HTML format.

Luca Rizzi, Francesco Zucconi

Global Kodaira-Spencer class and Massey products

📥 Full Text (PDF format), posted online 27 May 2024.

Abstract. We define a new notion of supported global deformation class for a semistable family of complex varieties over a curve f:X→B. We use this notion to study when X, possibly up to a finite covering, has a generically finite morphism onto a product B ×Y with Y of general type.

Tilak Bhattacharya

Some results for the Asymptotics and the Strong Minimum Principle for solutions to some nonlinear parabolic equations

📥 Full Text (PDF format), posted online 17 May 2024.

Abstract. We extend some results by Bhattacharya and Marazzi on strong minimum principle and asymptotics of positive viscosity solutions to a class of doubly nonlinear parabolic equations.

Hùng Việt Chu

Strong partially greedy bases with respect to an arbitrary sequence

📥 Full Text (PDF format), postedd online 13 May 2024.

Abstract. For Schauder bases, Dilworth et al. introduced and characterized the partially greedy property, which is strictly weaker than the (almost) greedy property. Later, Berasategui et al. defined and studied the strong partially greedy property for general bases. Let n be any strictly increasing sequence of positive integers. In this paper, we define the strong partially greedy property with respect to n, called the (n, strong partially greedy) property. We give characterizations of this new property, study relations among (n, strong partially greedy) properties for different sequences n, establish Lebesgue-type inequalities for the (n, strong partially greedy) parameter, investigate (n, strong partially greedy) bases with gaps, and weighted (n, strong partially greedy) bases, to name a few. Furthermore, we introduce the (n, almost greedy) property and equate the property to a strengthening of the (n, strong partially greedy) property. This paper can be viewed both as a survey of recent results regarding strong partially greedy bases and as an extension of these results to an arbitrary sequence instead of N.

Lucio Cadeddu, Antonio Greco, Benyam Mebrate

Non-autonomous overdetermined problems for the normalized p-Laplacian

📥 Full Text (PDF format), posted online 27 March 2024.

Abstract. We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition at the boundary. The definitions and statements needed to understand the main results are recalled in detail.

Lucio Boccardo, Andrea Dall'Aglio

Bounded solutions for Dirichlet problems with degenerate coercivity and a quadratic gradient term

📥 Full Text (PDF format), posted online 27 March 2024.

Abstract. We give existence results for weak solutions of Dirichlet problems for elliptic equations having degenerate coercivity and a first order term which has quadratic growth with respect to the gradient. The proof is based on the use of test functions having exponential growth.

Fatima Ezzahra Bourhim, Ali El Mfadel, M'hamed Elomari, Naoufel Hatime

Existence and uniqueness results for nonlinear hybrid Ψ-Caputo-type fractional differential equations with nonlocal periodic boundary conditions

📥 Full Text (PDF format), posted online 21 February 2024.

Abstract. In this paper, we consider a nonlinear fractional hybrid differential equation involving the Ψ-Caputo fractional operator with nonlocal periodic boundary conditions. Based on Lipschitz and Carathéodory conditions and via the Krasnoselskii fixed point theorem and some basic fractional analysis techniques, we discuss the existence and uniqueness of solutions to the proposed problem. Moreover, for a specific class of continuous functions, we prove some fundamental fractional differential inequalities. We finish this work with a non-trivial example.