Homogenization of nonconvex viscous Hamilton–Jacobi equations in stationary ergodic media in one dimension

Abstract We establish homogenization for nondegenerate viscous Hamilton–Jacobi equations in one space dimension when the diffusion coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> are general stationary ergodic processes in x . Our result is valid under mild regularity assumptions on a and H plus standard coercivity and growth assumptions (in p ) on the latter. In particular, we impose neither any additional condition on the law of the media nor any shape restriction on the graph of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">↦</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . Our approach consists of two main steps: (i) constructing a suitable candidate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="true">―</mml:mo> </mml:mover> </mml:mrow> </mml:math> for the effective Hamiltonian; (ii) proving homogenization. In the first step, we work with the set E of all points at which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="true">―</mml:mo> </mml:mover> </mml:mrow> </mml:math> is naturally determined by correctors with stationary derivatives. We prove that E is a closed subset of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> that is unbounded from above and below, and, if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>≠</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> , then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="true">―</mml:mo> </mml:mover> </mml:mrow> </mml:math> can be extended continuously to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> by setting it to be constant on each connected component of E c . In the second step, we use a key bridging lemma, comparison arguments and several general results to verify that homogenization holds with this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="true">―</mml:mo> </mml:mover> </mml:mrow> </mml:math> as the effective Hamiltonian.