王兵賢　徐梅　張玲萍

摘要：Robin系數(shù)在熱傳導(dǎo)模型中刻畫了熱傳導(dǎo)區(qū)域邊界上的熱交換，是一類非常重要的參數(shù)，本文基于某小時(shí)段溫度測量值反演熱傳導(dǎo)模型中的Robin系數(shù).首先，在邊界值以及測量值滿足一定的光滑性條件時(shí)，給出了反問題解的唯一性；其次，基于Tikhonov正則化思想，通過構(gòu)造目標(biāo)泛函將反問題轉(zhuǎn)化為求目標(biāo)泛函的極小值，并證明了泛函極小元的存在性.

關(guān)鍵詞：熱傳導(dǎo)方程；Robin系數(shù)；反問題；唯一性；極小元

中圖分類號：O 241.82文獻(xiàn)標(biāo)志碼：A文章編號：1001－988Ⅹ（2024）02－0026－03

Uniqueness of solution to inverse problem for the Robin coefficientin heat conduction equation and existence of its regularized solution

WANG Bing－xian，XU Mei，ZHANG Ling－ping

Abstract：The Robin coefficient characterizes the heat exchange on the edge of the heat conduction region in the heat conduction model，which is a very important parameter.This article discussed the inversion problem of the Robin coefficient in the heat conduction model based on temperature measurements during a certain period of time.Firstly，the uniqueness result of the solution to the inverse problem was given under certain conditions of boundary and measured values.Then，based on Tikhonovs regularization idea，the objective functional was constructed，and the inverse problem was transformed into finding the minimum of the objective functional，and the existence of minimizer was proved.

Key words：heat conduction equation;Robin coefficient;inverse problem;uniqueness;minimizer

0 引言

設(shè)區(qū)域ΩRd（d=2，3）為有界區(qū)域，且具有Lipchitz邊界Ω，考慮熱傳導(dǎo)方程初邊值問題

4 結(jié)束語

本文討論了Robin系數(shù)反演問題解的唯一性以及目標(biāo)最優(yōu)化問題極小元的存在性.對于反問題的條件穩(wěn)定性、目標(biāo)泛函最優(yōu)化下降算法的研究，以及數(shù)值模擬，我們將另文討論.

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（責(zé)任編輯 馬宇鴻）

收稿日期：2023－05－05;修改稿收到日期：2023－05－30

基金項(xiàng)目：國家自然科學(xué)基金資助項(xiàng)目（11501236）；江蘇省高校自然科學(xué)基金面上項(xiàng)目（18kJD110002）；淮陰師范學(xué)院博士啟動(dòng)基金項(xiàng)目（31WBX00）

作者簡介：王兵賢（1978—），男，甘肅民勤人，副教授，博士.主要研究方向?yàn)閿?shù)學(xué)物理反問題及統(tǒng)計(jì)模型快速算法.E－mail：wangbingxian@126.com