PDE - Chapter III - Section 3.3

Understanding the Space H10

Introduction to Density in Function Spaces

• The speaker introduces the concept of space H10, noting that while D is dense in L2, it is not dense in H1. This raises questions about how a larger space can contain a subset that isn't dense.

• The distinction between topologies of L2 and H1 is clarified; D's density in L2 pertains to its topology derived from the L2 norm, which differs from the topology of H1 due to non-equivalent norms.

Definition and Properties of H10

• H10 is defined as the pre-image of (0, 0) by the trace operator, indicating functions in H1 that equal zero at both ends.

• Key properties include:

• H10 being the closure of D under the norm of H1.

• Inclusion of H10 within H1 but not equating them; some functions exist in H1 that do not belong to H10.

• Establishing that with respect to its norm, H10 forms a Hilbert space.

Poincaré Inequality and Its Implications

• The speaker introduces Poincaré's inequality, stating it bounds the L2 norm of a function in H10 by a constant times the L2 norm of its derivative. This constant depends solely on the bounded interval considered.

• A contrast is drawn with functions in H1 where such bounding does not hold; for instance, constant functions have derivatives equal to zero yet can have unbounded norms.

Conceptual Analogy and Further Insights

• An analogy involving a "flying carpet" illustrates how without constraints (like boundary conditions), functions can take any value without being bound by their derivatives.

• In conclusion regarding Poincaré's inequality: if v belongs to H10, then its L2 norm will be bounded by a constant multiple of its derivative’s L2 norm.

Norm Characteristics in Function Spaces

• The discussion highlights how what typically serves as a semi-norm may become an actual norm within space H10 due to Poincaré's inequality.