Scalar Integral Functions Research Articles

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Lipschitz bounds for integral functionals with (p,q)-growth conditions

Abstract We study local regularity properties of local minimizers of scalar integral functionals of the form ℱ ⁢ [ u ] := ∫ Ω F ⁢ ( ∇ ⁡ u ) - f ⁢ u ⁢ d ⁢ x \mathcal{F}[u]:=\int_{\Omega}F(\nabla u)-fu\,dx where the convex integrand F satisfies controlled ( p , q ) {(p,q)} -growth conditions. We establish Lipschitz continuity under sharp assumptions on the forcing term f and improved assumptions on the growth conditions on F with respect to the existing literature. Along the way, we establish an L ∞ {L^{\infty}} - L 2 {L^{2}} -estimate for solutions of linear uniformly elliptic equations in divergence form, which is optimal with respect to the ellipticity ratio of the coefficients.

On the regularity of minimizers for scalar integral functionals with (p,q)-growth

We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called $(p,q)$-growth. In particular, we establish Lipschitz regularity under the condition $\frac{q}p<1+\frac{2}{n-1}$ for $n\geq3$ which improves a classical result due to Marcellini~[JDE'91].

A Harnack inequality for the quasi-minima of scalar integral functionals with general growth conditions

In this paper we proof a Harnack inequality and a regularity theorem for quasi-minima of scalar integral functionals with general growth conditions.

An Harnack inequality for the quasi-minima of scalar integral functionals with nearly linear growth conditions

In this paper we proof an Harnack inequality and a regularity theorem for quasi-minima of scalar integral functionals of the calculus of variations with nearly linear growth conditions. Particularly our results are applicable to functional with L Ln growth conditions.

$$L^{\Phi }-L^{\infty }$$ L Φ - L ∞ inequalities and new remarks on the Hölder continuity of the quasi-minima of scalar integral functionals with general growths

In this paper, we show a new regularity theorem for quasi-minima of scalar integral functionals of the calculus of variations with general growths. Let us consider functionals as the following one $$\begin{aligned} \ J\left[ u,\Omega \right] =\int \limits _{\Omega }f\left( x,u\left( x\right) , \nabla u\left( x\right) \right) \,\mathrm{d}x \end{aligned}$$ where $$f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^{N}\rightarrow {\mathbb {R}} $$ is a Caratheodory function that verifies the following hypothesis of growth $$\begin{aligned} c_{1}\Phi \left( \left| z\right| \right) -b\left( x\right) \left( \Phi \left( \left| s\right| \right) \right) ^{\gamma }-a\left( x\right) \le f\left( x,s,z\right) \le c_{2}\Phi \left( \left| z\right| \right) +b\left( x\right) \left( \Phi \left( \left| s\right| \right) \right) ^{\gamma }+a\left( x\right) \end{aligned}$$ for each $$z\in {\mathbb {R}}^{N}$$ , $$s\in {\mathbb {R}} $$ and for $${\mathcal {L}}^{N}$$ -a. e. $$x\in \Omega $$ . Moreover, $$\Phi $$ is a N-function, $$c_{1}$$ and $$c_{2}$$ are two positive real constants, with $$ c_{1}<c_{2}$$ , $$\Omega \subset {\mathbb {R}}^{N}$$ is an open subset, $$N\ge 2$$ , $$\ b\left( x\right) \in L_{\mathrm{loc}}^{r}\left( \Omega \right) $$ , $$a\left( x\right) \in L_{\mathrm{loc}}^{t}\left( \Omega \right) $$ with $$t=\frac{N}{1-N\epsilon }$$ , $$r=\frac{N}{N-\gamma \left( N-1\right) -N\epsilon }$$ , $$1<\gamma <\frac{N}{N-1}$$ , $$0<\epsilon <\frac{1}{N}$$ and $$ b\left( x\right) ,a\left( x\right) \ge 0$$ on $$\Omega $$ . The functional $$\ J \left[ u,\Omega \right] $$ is defined on the Orlicz–Sobolev space $$ W_{0}^{1}L^{\Phi }\left( \Omega \right) +g$$ where $$g\in W^{1}L^{\Phi }\left( \Omega \right) $$ , $$\Phi $$ is a N-function. Moreover, we suppose that $$\Phi \in \triangle _{2}$$ and we will give some further hypotheses on the N-function $$\Phi $$ .

Remarks on the continuity of the local minimizer of scalar integral functionals with nonstandard growth conditions

Remarks on the continuity of the local minimizer of scalar integral functionals with nonstandard growth conditions

Boundedness of local minimizers of anisotropic scalar integral functionals with general growth conditions

Boundedness of local minimizers of anisotropic scalar integral functionals with general growth conditions

Remarks on the Continuity of the Local Minimizer of Scalar Integral Functionals With Nonstandard General Growth Conditions

In this paper we show a regularity theorem for local minima of scalar integral functionals of the Calculus of Variations with nonstandard general growth conditions. Let us consider functionals in the following form\begin{equation*}\mathcal{F}\left[ u,\Omega \right] =\int\limits_{\Omega }f\left( x,u\left(x\right) ,\nabla u\left( x\right) \right),dx\end{equation*}where $f$: $\Omega \times\mathbb{R} \times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a Carath\'{e}odory function\ satisfying the inequalities\begin{equation*}\Phi \left( \left\vert z\right\vert \right) -c_{1}\leq f\left( x,s,z\right)\leq c_{2}\left[ 1+\left( \Phi ^{\ast }\left( \left\vert z\right\vert\right) \right) ^{\beta }+\left( \Phi ^{\ast }\left( \left\vert s\right\vert\right) \right) ^{\beta }\right]\end{equation*}for each $z\in\mathbb{R}^{N}$, $s\in\mathbb{R}$ and for $\mathcal{L}^{N}$-a. e. $x\in \Omega $, where $c_{1}$ and $c_{2}$ are two positive real constants, with $c_{1}<c_{2}$, $\Omega $ is an open subset of $\mathbb{R}^{N}$, $N\geq 2$, $\Phi \in \triangle _{2}^{m}\cap \nabla _{2}^{r}$ [Definition 6 and Definition 8], $1\leq r<m<N$ and the function $\Phi ^{\ast}$ is the Sobolev conjugate of $\Phi $ [Definition 12], $\beta $ is a positive real number that we will opportunely fix [Hypothesis $H_{1,f}$].

Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Functionals with General Growth Conditions

In this paper we proof a Harnack inequality and a regularity theorem for local-minima of scalar intagral functionals with general growth conditions.

Obtaining one-loop gravity amplitudes using spurious singularities

The decomposition of a one-loop scattering amplitude into elementary functions with rational coefficients introduces spurious singularities which afflict individual coefficients but cancel in the complete amplitude. These cancellations create a web of interactions between the various terms. We explore the extent to which entire one-loop amplitudes can be determined from these relationships starting with a relatively small input of initial information, typically the coefficients of the scalar integral functions as these are readily determined. In the context of one-loop gravity amplitudes, of which relatively little is known, we find that some amplitudes with a small number of legs can be completely determined from their box coefficients. For increasing numbers of legs, ambiguities appear which can be determined from the physical singularity structure of the amplitude. We illustrate this with the four-point and N=1,4 five-point (super)gravity one-loop amplitudes.

Simplicity in the structure of QED and gravity amplitudes

We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED `no-triangle' property is true for all helicity configurations and has similarities to the `no-triangle' property found in the case of maximal N = 8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude.

Absence of triangles in maximal supergravity amplitudes

From general arguments, we show that one-loop n-point amplitudes in colourless theories satisfy a new type of reduction formula. These lead to the existence of cancellations beyond supersymmetry. Using such reduction relations we prove the no-triangle hypothesis in maximal supergravity by showing that in four dimensions the n-point graviton amplitude contain only scalar box integral functions. We also discuss the reduction formulas in the context of gravity amplitudes with less and no supersymmetry.

The no-triangle hypothesis for Script N = 8 supergravity

We study the perturbative expansion of N=8 supergravity in four dimensions from the viewpoint of the ``no-triangle'' hypothesis, which states that one-loop graviton amplitudes in N=8 supergravity only contain scalar box integral functions. Our computations constitute a direct proof at six-points and support the no-triangle conjecture for seven-point amplitudes and beyond.

Perturbative Gravity and Twistor Space

The recent progress in computing gauge theory amplitudes can be extended, in many cases, to theories incorporating gravity. This has improved our understanding of the perturbative expansion of N=8 supergravity supporting the ``no-triangle hypothesis'' that N=8 one-loop amplitudes may be expressed in terms of scalar box integral functions.

Six-point one-loop [formula omitted] supergravity NMHV amplitudes and their IR behaviour

We present compact formulas for the box coefficients of the six-graviton NMHV one-loop amplitudes in N=8 supergravity. We explicitly demonstrate that the corresponding box integral functions, with these coefficients, have the complete IR singularities expected of the one-loop amplitude. This is strong evidence for the conjecture that N=8 one-loop amplitudes may be expressed in terms of scalar box integral functions. This structure, although unexpected from a power counting viewpoint, is analogous to the structure of N=4 super-Yang–Mills amplitudes. The box-coefficients match the tree amplitude terms arising from recursion relations.

Two-loop four-gluon amplitudes in N = 4 super-Yang-Mills

Using cutting techniques we obtain the two-loop N = 4 super-Yang-Mills helicity amplitudes for four-gluon scattering in terms of scalar integral functions. The N = 4 amplitudes are considerably simpler than corresponding QCD amplitudes and therefore provide a testing ground for exploring two-loop amplitudes. The amplitudes are constructed directly in terms of gauge invariant quantities and therefore remain relatively compact throughout the calculation. We also present a conjecture for the leading color four-gluon amplitudes to all orders in the perturbative expansion.

Genericity and existence of a minimum for scalar integral functionals

This paper is concerned with the existence of a minimum in a Sobolev space for the functional $$F_g (x): = \int_0^T {g(x(t)){\text{ }}dt + \int_0^T {h(x'(t)){\text{ }}dt,{\text{ }}x(0) = a,{\text{ ...

Genericity and existence of a minimum for scalar integral functionals

This paper is concerned with the existence of a minimum in a Sobolev space for the functional $$F_g (x): = \int_0^T {g(x(t)){\text{ }}dt + \int_0^T {h(x'(t)){\text{ }}dt,{\text{ }}x(0) = a,{\text{ }}x(T) = b,} }$$ wherea,b are real numbers,g is a continuous map, andh is lower semicontinuous, satisfying adequate growth conditions.