What does this research mean for the field?

Viscosity solutions exist for a broad class of first-order partial differential equations with merely measurable coefficients and superlinear gradient growth, without requiring continuity assumptions. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

June 19, 2026Open Access

Viscosity Solutions for First‐Order PDEs With Measurable Coefficients and Superlinear Gradient Growth

Key Points

This study aims to establish a general existence theorem for viscosity solutions to a broad class of first-order PDEs.
Developed a general existence theorem applicable to partial differential equations with measurable coefficients.
Derived propositions for specific families of PDEs to demonstrate the flexibility of the existence results.
Validated the existence theorem for PDEs with measurable coefficients without continuity assumptions.
Provided examples that extend beyond previous theorems requiring stronger regularity hypotheses.

Abstract

Inspired by a fundamental existence result in the theory of PDEs, we present a general existence theorem for viscosity solutions in the standard sense that is applicable to a wide class of partial differential equations. These equations are characterized by coefficients that are merely measurable, with no continuity assumptions imposed. To illustrate the scope and flexibility of the main result, we derive several propositions for specific families of PDEs, including examples that fall outside the reach of prior existence theorems with stronger regularity hypotheses.

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Cite This Study

Hosseini et al. (Thu,) studied this question.

synapsesocial.com/papers/6a34dd4965a5b0777af2d25f https://doi.org/https://doi.org/10.1155/jom/6026158

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