By Lucia M.

We think about the equationIf Ω is of sophistication , we exhibit that this challenge has a non-trivial resolution u λ for every λ ∊ (8π, λ*). the price λ* depends upon the area and is bounded from lower than by way of 2 j zero 2 π, the place j zero is the 1st 0 of the Bessel functionality of the 1st form of order 0 (λ*≥ 2 j zero 2 π > eight π). additionally, the family members of answer u λ blows-up as λ → eight π.

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Preimages If ƒ: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is the set of all elements of X that map to y: The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f. Similarly, if S is any subset of Y, the preimage of S is the set of all elements of X that map to S: For example, take a function ƒ: R → R, where ƒ: x → x2. This function is not invertible for reasons discussed above. Yet preimages may be defined for subsets of the codomain: The preimage of a single element y ∈ Y – a singleton set {y} – is sometimes called the fiber of y.

Facts about continuous functions If two functions f and g are continuous, then f + g, fg, and f/g are continuous. (Note. The only possible points x of discontinuity of f/g are the solutions of the equation g(x) = 0; but then any such x does not belong to the domain of the function f/g. ) The composition f o g of two continuous functions is continuous. If a function is differentiable at some point c of its domain, then it is also continuous at c. The converse is not true: a function that is continuous at c need not be differentiable there.

If the function ƒ is differentiable, then the inverse ƒ−1 will be differentiable as long as ƒ′(x) ≠ 0. The derivative of the inverse is given by the inverse function theorem: If we set x = ƒ–1(y), then the formula above can be written This result follows from the chain rule. The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable function ƒ: Rn → Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of ƒ at p is invertible.