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I have e.g. the following DensityPlot, where the PlotRange exceeds the internal DataRange:

DensityPlot[{-x^2 - y^2, -x^2 - y^2}, {x, -1, 1}, {y, -1, 1}, ColorFunction -> "Rainbow", 
 PlotLegends -> Automatic, PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}}]

enter image description here

How can I fill the white frame around the density plot with the color corresponding to lowest legend value (here -2)? It can be any color scheme.

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You can also use options epilog and prolog to combine two Graphics objects:

DensityPlot[{-x^2 - y^2, -x^2 - y^2}, {x, -1, 1}, {y, -1, 1}, 
ColorFunction -> "Rainbow", PlotLegends -> Automatic, Frame -> True,
PlotRange -> {{-1.5, 1.5},{-1.5, 1.5}},
Prolog -> {{ColorData["Rainbow"]@0,Rectangle[{-1.4, -1.4}, {1.4, 1.4}]}}]

Density plot

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You can combine two DensityPlots

xymin = -2;
Show[DensityPlot[xymin, {x, -1.5, 1.5}, {y, -1.5, 1.5}, ColorFunction -> "Rainbow"],
     DensityPlot[-x^2 - y^2, {x, -1, 1}, {y, -1, 1}, ColorFunction -> "Rainbow",
                 PlotLegends -> Automatic, PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}}]]

enter image description here

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