This chapter focuses on the strong unique continuation property for second-order elliptic equations with real coefficients in the principal part. The analysis begins with the essential concept of three-sphere inequalities, which play a crucial role in establishing this property. These inequalities are introduced alongside doubling inequalities, both derived using Carleman estimates applied to elliptic operators. The chapter explores the derivation of Carleman estimates for the Laplace operator and extends the approach to equations with variable coefficients, introducing technical challenges such as the use of geodesic polar coordinates.

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Optimal Three Sphere and Doubling Inequality for Second-Order Elliptic Equations

  • Sergio Vessella

摘要

This chapter focuses on the strong unique continuation property for second-order elliptic equations with real coefficients in the principal part. The analysis begins with the essential concept of three-sphere inequalities, which play a crucial role in establishing this property. These inequalities are introduced alongside doubling inequalities, both derived using Carleman estimates applied to elliptic operators. The chapter explores the derivation of Carleman estimates for the Laplace operator and extends the approach to equations with variable coefficients, introducing technical challenges such as the use of geodesic polar coordinates.