Regularity for scalar integrals without structure conditions

Integrals of the Calculus of Variations with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:math> {p,q} -growth may have not smooth minimizers, not even bounded, for general <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:math> {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {f(x,ξ)} with dependence on the modulus of the gradient, i.e. <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {f(x,ξ)=g(x,|ξ|)} . Without imposing structure conditions, we prove that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mi>q</m:mi> <m:mi>p</m:mi> </m:mfrac> </m:math> {q/p} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.