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We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$ . Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

The teleparallel equivalent of higher order Lagrangians like $L_{\Box R}=-R+a_{0}R^{2}+a_{1}R\Box R$ can be obtained by means of the boundary term $B=2\nabla_{\mu}T^{\mu}$. In this perspective, we derive the field equations in presence of matter for higher-order teleparallel gravity considering, in particular, sixth-order theories where the $\Box$ operator is linearly included. In the weak field approximation, gravitational wave solutions for these theories are derived. Three states of polarization are found: the two standard $+$ and $\times$ polarizations, namely 2-helicity massless transverse tensor polarizations, and a 0-helicity massive, with partly transverse and partly longitudinal scalar polarization. Moreover, these gravitational waves exhibit four oscillation modes related to four degrees of freedom: the two classical $+$ and $\times$ tensor modes of frequency $\omega_{1}$, related to the standard Einstein waves with $k^{2}_{1}=0$; two mixed longitudinal-transverse scalar modes for each frequencies $\omega_{2}$ and $\omega_{3}$, related to two different 4-wave vectors, $k^{2}_{2}=M_{2}^{2}$ and $k^{2}_{3}=M^{2}_{3}$. The four degrees of freedom are the amplitudes of each individual mode, i.e. $\hat{\epsilon}^{(+)}\left(\omega_{1}\right)$, $\hat{\epsilon}^{(\times)}\left(\omega_{1}\right)$, $\hat{B}_{2}\left(\mathbf{k}\right)$, and $\hat{B}_{3}\left(\mathbf{k}\right)$., Comment: 27 pages, accepted for publication in Classical and Quantum Gravity

In this paper we study minimal realizations in L p ( R N ) of the second order elliptic operator A b , c : = ( 1 + | x | α ) Δ + b | x | α − 2 x ⋅ ∇ − c | x | α − 2 − | x | β , x ∈ R N , where N ≥ 3 , α ∈ [ 0 , 2 ) , β > 0 , and b , c are real numbers. We use quadratic form methods to prove that ( A b , c , C c ∞ ( R N ∖ { 0 } ) ) admits an extension that generates an analytic C 0 -semigroup for all p ∈ ( 1 , ∞ ) . Moreover, we give conditions on the coefficients under which this extension is precisely the closure of ( A b , c , C c ∞ ( R N ∖ { 0 } ) ) .

We derive the weak field limit for a gravitational Lagrangian density $L_{g}=(R+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$ where higher-order derivative terms in the Ricci scalar $R$ are taken into account. The interest for this kind of effective theories comes out from the consideration of the infrared and ultraviolet behaviors of gravitational field and, in general, from the formulation of quantum field theory in curved spacetimes. Here, we obtain solutions in weak field regime both in vacuum and in the presence of matter and derive gravitational waves considering the contribution of $R\Box^{k}R$ terms. By using a suitable set of coefficients $a_{k}$, it is possible to find up to $(p+2)$ normal modes of oscillation with six polarization states with helicity 0 or 2. Here $p$ is the higher order term in the $\Box$ operator appearing in the gravitational Lagrangian. More specifically: the mode $\omega_{1}$, with $k^{2}=0$, has transverse polarizations $\epsilon_{\mu\nu}^{\left(+\right)}$ and $\epsilon_{\mu\nu}^{\left(\times\right)}$ with helicity 2; the $(p+1)$ modes $\omega_{m}$, with $k^{2}\neq0$, have transverse polarizations $\epsilon_{\mu\nu}^{\left(1\right)}$ and non-transverse ones $\epsilon_{\mu\nu}^{\left(\text{TT}\right)}$, $\epsilon_{\mu\nu}^{\left(\text{TS}\right)}$, $\epsilon_{\mu\nu}^{\left(L\right)}$ with helicity 0., Comment: 21 pages, to appear in Int.J.Geom.Meth.Mod.Phys

We derive the gravitational waves for $f\left(T, B\right)$ gravity, an extension of teleparallel gravity containing the torsion scalar $T$ and the boundary term $B$, and demonstrate that it is equivalent to $f(R)$ gravity. Gravitational waves, in this theory, show three polarizations: the two standard, plus and cross, of General Relativity, which are purely transverse, massless tensor polarization modes with two-helicity, and an additional massive scalar mode with zero-helicity. The last one is a mix of a longitudinal mode and a transverse breathing mode. The boundary term $B$ excites the extra scalar polarization and the mass of scalar field breaks the symmetry of the TT gauge by adding a new degree of freedom, namely a single mixed scalar polarization., Comment: 14 pages, accepted for publication in Eur. Phys. J. C

In this paper we prove some weighted \(W^{2,2}\)-a priori bounds for a class of linear, elliptic, second-order, differential operators of Cordes type in certain weighted Sobolev spaces on unbounded open sets \(\varOmega \) of \(\mathbb {R}^{n},\,n\ge 2\). More precisely, we assume that the leading coefficients of our differential operator satisfy the so-called Cordes type condition, which corresponds to uniform ellipticity if \(n=2\) and implies it if \(n\ge 3\), while the lower order terms are in specific Morrey type spaces. Here, our analytic technique mainly makes use of the existence of a topological isomorphism from our weighted Sobolev space, denoted by \(W^{2,2}_s(\varOmega )\) (\(s\in \small \mathbb {R}\)), whose weight is a suitable function of class \(C^2(\bar{\varOmega })\), to the classical Sobolev space \(W^{2,2}(\varOmega )\), which allow us to exploit some well-known unweighted a priori estimates. Using the above mentioned \(W^{2,2}_s\)-a priori bounds, we also deduce some existence and uniqueness results for the related Dirichlet problems in the weighted framework.

We give an overview on some results concerning the unique solvability of the Dirichlet problem inW2,p,p>1, for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case.

In this paper we prove some a priori bounds for the solutions of the Dirichlet problem for elliptic equations with singular coefficients in weighted Sobolev spaces.

In this paper we prove an existence and uniqueness theorem for the Dirichlet problem for linear elliptic equations (in unbounded domains) whose leading coefficients are locally VMO and satisfy a suitable condition at infinity.

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝn, n ≥ 2, not necessarily bounded or regular.

This paper is a continuation of [4] and extends the well-known theorem of C. Miranda (Ann. Mat. Pura Appl. (4) 61 (1963)) concerning a priori estimates of weak solutions of linear elliptic equations with discontinuous coefficients in a bounded open subset of Rn . In [4] we derive such estimates in weighted spaces considered over an arbitrary open subset of Rn . In this paper we obtain the same result replacing a condition on the coefficients by a considerably weaker one. Mathematics subject classification (2000): 35J25, 46E35.

We prove an existence and uniqueness result for the Dirichlet problem for a class of elliptic equations with singular data in weighted Sobolev spaces.

We deepen the study of some Morrey type spaces, denoted byMp,λ(Ω), defined on an unbounded open subsetΩofℝn. In particular, we construct decompositions for functions belonging to two different subspaces ofMp,λ(Ω), which allow us to prove a compactness result for an operator in Sobolev spaces. We also introduce a weighted Morrey type space, settled between the above-mentioned subspaces.

In this paper we prove some interpolation inequalities between functions and their derivatives in the class of weighted Sobolev spaces defined on unbounded open subset Ω ⊂ R n .

Some estimates for solutions of the Dirichlet problem for second-order elliptic equations are obtained in this paper. Here the leading coefficients are locally VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function.

We prove some weighted $W^{2,2}$ -a priori bounds for a class of elliptic second order linear differential operators of Cordes type on unbounded domains of $\mathbb {R}^{n}$ , $n\geq2$ .