The best approximation of a given function in L ²-norm by Lipschitz functions with gradient constraint

The starting point of this paper is the study of the asymptotic behavior, as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {p→∞} , of the following minimization problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>min</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo maxsize="210%" minsize="210%">{</m:mo> <m:mrow> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mi>p</m:mi> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>v</m:mi> <m:mo>-</m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace="12.5pt">,</m:mo> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize="210%" minsize="210%">}</m:mo> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> min{\{}1/p∫Ω|∇ v|ᵖ+1/2∫Ω(% v-f)², v∈ W1,p(Ω){\}}. We show that the limit problem provides the best approximation, in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L²} -norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(N-1)} -Hausdorff measure of the jump set of the function.